Plane and solid geometry - triangles, circles, polygons, and classical geometric arguments.
These practice problems are analytic and vector geometry. You will compute with vectors in space, write equations of lines, planes, and spheres, and identify the standard quadric surfaces. The work is algebraic: components, products, and coordinate changes, with a geometric conclusion at the end. Each solution shows the vector calculation and then says what it means for the figure.
Vector Algebra, Scalar Product, and Vector Product build the toolkit: components, length, projection, orthogonality, and the area of a parallelogram. Triple Products and Volume turns that toolkit into volumes of parallelepipeds and tetrahedra. Conics and Transformations treats parabolas, ellipses, and hyperbolas, and the coordinate changes that simplify their equations. Planes, Lines, Spheres, Cylinders and Cones, and Quadric Surfaces are the solid-geometry chapters, from a single plane up to the standard quadrics.
Vector algebra is the easy entry; quadric surfaces are the hard end, so the difficulty control is worth using. Complex Numbers is a lighter companion if you want arithmetic in the plane before vectors. Applications of Definite Integrals computes volumes of some of the same solids by integration. Physics uses vectors for motion, and it does not replace the geometry practiced here.
Problem 1
HardVector Algebra
Given a square $ABCD$. Let $A_1, B_1, C_1, D_1$ be the midpoints of the sides of the square. Express in terms of vectors $\overrightarrow{AB}$ and $\overrightarrow{AD}$ the vectors $\overrightarrow{DA},\ \overrightarrow{BC},\ \overrightarrow{AC},\ \overrightarrow{BD},\ \overrightarrow{A_1B_1},\ \overrightarrow{B_1C_1},\ \overrightarrow{C_1D_1},\ \overrightarrow{A_1C}$. Calculate their magnitudes if $AB = 2$.
What is the mutual position of vectors $\vec{a}$ and $\vec{b}$ if: $1^\circ\quad |\vec{a}| + |\vec{b}| = |\vec{a} + \vec{b}| \qquad\qquad 2^\circ\quad |\vec{a}| - |\vec{b}| = |\vec{a} - \vec{b}|$.
What must be the mutual position of vectors $\vec{a}$ and $\vec{b}$ so that: $1^\circ\quad |\vec{a}+\vec{b}|=|\vec{a}-\vec{b}| \qquad\quad 2^\circ\quad |\vec{a}|+|\vec{b}|=|\vec{a}-\vec{b}|$ $3^\circ\quad |\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}|$.
Verify analytically and graphically the following vector equalities: $1^\circ\quad \vec{a}+\tfrac{1}{2}(\vec{b}-\vec{a})=\tfrac{1}{2}(\vec{a}+\vec{b}) \qquad\quad 2^\circ\quad \vec{a}-\tfrac{1}{2}(\vec{a}+\vec{b})=\tfrac{1}{2}(\vec{a}-\vec{b})$ $3^\circ\quad \frac{\vec{a}+\vec{b}}{2}-\frac{\vec{a}-\vec{b}}{2}=\vec{b}$.
What must be the mutual position and properties of vectors $\vec{a}$ and $\vec{b}$ so that: $1^\circ\quad \frac{\vec{a}}{|\vec{a}|}=\frac{\vec{b}}{|\vec{b}|} \qquad\quad 2^\circ\quad \vec{a}=-2\lambda\vec{b} \qquad\quad 3^\circ\quad \vec{a}+\vec{b}=\lambda(\vec{a}-\vec{b})$.
In a plane, points $A(0,-2)$, $B(4,2)$, $C(4,-2)$ are given. Forces $\overrightarrow{OA}$, $\overrightarrow{OB}$, $\overrightarrow{OC}$ are drawn from the origin. Construct the resultant $\overrightarrow{OM}$ and find its projections on the coordinate axes. Express forces $\overrightarrow{OA}$, $\overrightarrow{OB}$, $\overrightarrow{OC}$ and $\overrightarrow{OM}$ in terms of unit vectors $\vec{i}$ and $\vec{j}$.
In an isosceles trapezoid $OACB$: $\angle BOA = 60^\circ$, $\overline{OB} = \overline{BC} = \overline{CA} = 2$, and $M$, $N$, $P$ are midpoints of sides $BC$, $AC$ and $OA$ respectively. Express vectors $\overrightarrow{AC}$, $\overrightarrow{OM}$, $\overrightarrow{ON}$, $\overrightarrow{MN}$, $\overrightarrow{PM}$ and $\overrightarrow{NP}$ in terms of unit vectors $\vec{m}$ and $\vec{n}$ of sides $\overrightarrow{OA}$ and $\overrightarrow{OB}$.
A cube with edge length $2$ cm has vertices $ABCD$ and $A_1B_1C_1D_1$. Let $M$ and $N$ be the centers of faces $A_1B_1C_1D_1$ and $BCC_1B_1$ respectively. Express vectors $\overrightarrow{AM}$, $\overrightarrow{AN}$, $\overrightarrow{MN}$ and $\overrightarrow{CA_1}$ in terms of unit vectors $\vec{i}$, $\vec{j}$, $\vec{k}$ of directions $\overrightarrow{AB}$, $\overrightarrow{AD}$ and $\overrightarrow{AA_1}$. Show that $\overrightarrow{MN}$ is parallel to $\overrightarrow{A_1B}$.
Let $N$ and $M$ be midpoints of sides $OA$ and $AB$ of triangle $OAB$. Express analytically and decompose geometrically: $1^\circ$ vectors $\overrightarrow{OM}$ and $\overrightarrow{MN}$ in terms of $\overrightarrow{OA}=\vec{a}$ and $\overrightarrow{OB}=\vec{b}$; $2^\circ$ vectors $\overrightarrow{ON}$ and $\overrightarrow{BN}$ in terms of $\vec{b}$ and $\overrightarrow{OM}=\vec{c}$.
Let $\vec{i}$ and $\vec{j}$ be unit vectors of the coordinate axes. Construct vectors $\vec{a} = 4\vec{i}+\vec{j}$ and $\vec{b} = \vec{i}+3\vec{j}$. What are the projections of $\vec{a}$ and $\vec{b}$ onto the coordinate axes?
Determine $\alpha$ and $\beta$ from the vector equality $\vec{c} = \alpha\vec{a}+\beta\vec{b}$, where $\vec{a} = 4\vec{i}+\vec{j}$, $\vec{b} = -2\vec{i}+3\vec{j}$, $\vec{c} = \vec{i}+4\vec{j}$, and $\vec{i}$, $\vec{j}$ are unit vectors of a rectangular coordinate system. Then decompose $\vec{a}$ analytically and geometrically in terms of $\vec{b}$ and $\vec{c}$.
If two vectors $\vec{a}$ and $\vec{b}$ are collinear, one can always find a constant $\lambda$ such that $\vec{a} = \lambda\vec{b}$, or two constants $\alpha$ and $\beta$ such that $\alpha\vec{a}+\beta\vec{b} = \vec{0}$. What is the relation between the constants $\lambda$, $\alpha$ and $\beta$? Determine $\alpha$ if $\lambda = 1$, $\beta = 2$.
Can vectors $\vec{a}=\vec{i}-3\vec{j}$, $\vec{b}=\alpha\vec{i}+\vec{j}$ and $\vec{c}=\vec{i}+\beta\vec{j}$ all be collinear ($\alpha$ and $\beta$ are parameters)? When will $\vec{b}$ and $\vec{c}$ be collinear independently of $\vec{a}$?
A regular hexagon $OABCDE$ has side $OA = 3$. Given $\overrightarrow{OA}=\vec{m}$, $\overrightarrow{AB}=\vec{n}$ and $\overrightarrow{BC}=\vec{p}$, find the relations between vectors $\vec{m}$, $\vec{n}$ and $\vec{p}$, and express vectors $\overrightarrow{OD}$, $\overrightarrow{DA}$, $\overrightarrow{EB}$ and $\overrightarrow{OC}$ in terms of them.
Triangle $ABC$ is constructed on vectors $\overrightarrow{BC}=\vec{a}$ and $\overrightarrow{CA}=\vec{b}$. a) Express the median vectors $\overrightarrow{AA_1}$, $\overrightarrow{BB_1}$ and $\overrightarrow{CC_1}$ in terms of $\vec{a}$ and $\vec{b}$. b) Show that the three medians can form a triangle.
Point $B$ divides the arc $\overset{\frown}{AC} = 90^\circ$ of a circle with center $O$ in the ratio $1:2$. Express analytically and geometrically the vector $\overrightarrow{OC} = \vec{c}$ in terms of vectors $\overrightarrow{OA} = \vec{a}$ and $\overrightarrow{OB} = \vec{b}$.
In triangle $ABC$, side $BC$ is divided by point $D$ in the ratio $m:n$. Express vector $\overrightarrow{AD}$ in terms of vectors $\overrightarrow{AB}=\vec{c}$ and $\overrightarrow{AC}=\vec{b}$.
If three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ are coplanar (lie in one plane or are parallel to one plane), then there exist constants $\alpha$ and $\beta$ such that $\vec{c} = \alpha\vec{a}+\beta\vec{b}$. The converse also holds. Prove both directions.
Given vectors $\vec{a}=\alpha\vec{i}+\vec{j}+4\vec{k}$, $\vec{b}=\vec{i}-2\alpha\vec{j}$ and $\vec{c}=3\alpha\vec{i}-3\vec{j}+4\vec{k}$, where $\vec{i}$, $\vec{j}$ and $\vec{k}$ are non-coplanar (linearly independent) unit vectors. Determine the scalar $\alpha$ so that the vectors are coplanar.
Given vectors $\vec{p}=2\vec{i}-3\vec{j}$, $\vec{q}=-\vec{i}+2\vec{j}$ and $\vec{a}=9\vec{i}+4\vec{j}$, where $\vec{i}$ and $\vec{j}$ are mutually orthogonal unit vectors. Decompose analytically and geometrically vector $\vec{a}$ along the directions of vectors $\vec{p}$ and $\vec{q}$.
Given vectors $\vec{a}=3\vec{p}-2\vec{q}$, $\vec{b}=-2\vec{p}+\vec{q}$ and $\vec{c}=7\vec{p}-4\vec{q}$, where $\vec{p}$ and $\vec{q}$ are unit vectors with $\angle(\vec{p},\vec{q})=\dfrac{\pi}{3}$. Decompose vector $\vec{a}$ along the directions of vectors $\vec{b}$ and $\vec{c}$ (analytically and geometrically).
Given vectors $\vec{p}=\vec{a}+\vec{b}+\vec{c}$, $\vec{a}=3\vec{m}-\vec{n}$, $\vec{b}=\vec{m}-2\vec{n}$ and $\vec{c}=-\vec{m}+7\vec{n}$. Find the linear dependence relation between vectors $\vec{p}$, $\vec{a}$ and $\vec{b}$.
Let $\vec{i}$, $\vec{j}$ and $\vec{k}$ be unit vectors of the coordinate axes of a spatial coordinate system. Decompose vector $\overrightarrow{OM}$ in terms of $\vec{i}$, $\vec{j}$ and $\vec{k}$ analytically and geometrically, where $M(x,y,z)$ is an arbitrary point in space and $O$ is the origin.
Given two points in space $M_1(x_1,y_1,z_1)$ and $M_2(x_2,y_2,z_2)$. Prove that $\overrightarrow{M_1M_2} = (x_2-x_1)\vec{i}+(y_2-y_1)\vec{j}+(z_2-z_1)\vec{k}$ and that $|\overrightarrow{M_1M_2}| = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$.
If $\alpha$, $\beta$ and $\gamma$ are the angles that vector $\overrightarrow{OM}$ makes with the coordinate axes, prove that $\cos\alpha = \frac{x}{\sqrt{x^2+y^2+z^2}}, \quad \cos\beta = \frac{y}{\sqrt{x^2+y^2+z^2}}, \quad \cos\gamma = \frac{z}{\sqrt{x^2+y^2+z^2}}$ and that $\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1$, where $x$, $y$, $z$ are the coordinates of point $M$.
Vector $\vec{r}$ makes angles $\alpha=30^\circ$ and $\beta=60^\circ$ with coordinate axes $Ox$ and $Oy$ respectively. Find the angle $\gamma$ that the vector makes with the $Oz$ axis.
Determine the position vector of point $M$ given that it makes an angle of $45^\circ$ with the $Ox$ axis, an angle of $60^\circ$ with the $Oy$ axis, its magnitude is $6$, and its $z$-coordinate is negative.
From point $A(2,1,-1)$ acts a force $\vec{F}$ with magnitude $7$ and projections: on the $x$-axis $X=2$, on the $y$-axis $Y=-3$. Determine the direction and sense of force $\vec{F}$.
Find points on the sphere with center at the origin and radius $R=2\sqrt{3}$, whose radius vector makes equal angles with the positive directions of the coordinate axes. What are those angles?
Three consecutive vertices of parallelogram $ABCD$ are given: $A(1,-2,0)$, $B(2,1,3)$, $C(-2,0,5)$. Find the fourth vertex $D$ and the length of diagonal $BD$.
Points $A(0,1,2)$, $B(2,0,1)$ and $C(3,1,0)$ are three consecutive vertices of a parallelogram. Find point $O'$ — the intersection of the diagonals — and the centroid of triangle $ABC$.
Three consecutive vertices of parallelogram $ABCD$: $A(-3,2,\lambda)$, $B(3,-3,1)$, $C(5,\lambda,2)$. $1^\circ$ Find the fourth vertex $D$. $2^\circ$ Find $\lambda$ such that $|\overrightarrow{AD}|=\sqrt{14}$. $3^\circ$ For the larger value of $\lambda$ found in $2^\circ$, find the linear dependence among $\overrightarrow{AD}$, $\overrightarrow{BD}$ and $\overrightarrow{AC}$, and decompose $\overrightarrow{AC}$ in terms of $\overrightarrow{AD}$ and $\overrightarrow{BD}$.
$1^\circ$ Prove that the scalar product satisfies the commutative law: $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$. $2^\circ$ Prove that the scalar product satisfies the distributive law: $\vec{a}\cdot(\vec{b}+\vec{c})=\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$.
$1^\circ$ Find all scalar products of unit coordinate vectors (orths) in space. $2^\circ$ Find $(\vec{i}-\vec{j}+\vec{k})\cdot(2\vec{i}-3\vec{j}-\vec{k})$.
Find the scalar product of vectors $\vec{a}=2\vec{m}-\vec{n}$ and $\vec{b}=\vec{m}-2\vec{n}$, where $|\vec{m}|=2$, $|\vec{n}|=4$ and $\angle(\vec{m},\vec{n})=\dfrac{\pi}{3}$.
Determine the parameter $\lambda$ so that vectors $\vec{a}\{2,\lambda,\lambda-1\}$ and $\vec{b}\{\lambda,\lambda-2,0\}$ have equal magnitudes, then find the angle between them.
If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ and the magnitudes $|\vec{a}|$, $|\vec{b}|$, $|\vec{c}|$ are known, find $\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}+\vec{b}\cdot\vec{c}$.
Find the projection of the vector $\vec{a}=2\vec{p}-3\vec{q}$ onto $\vec{b}=\vec{p}+\vec{q}$, given $|\vec p|=2$, $|\vec q|=3$ and $\angle(\vec p,\vec q)=60^\circ$.
A parallelogram is built on the vectors $\vec a=2\vec m+\vec n$ and $\vec b=\vec m-2\vec n$, where $\vec m$ and $\vec n$ are unit vectors with $\angle(\vec m,\vec n)=60^\circ$. Find the lengths of the diagonals and the angle between them.
In the cube $ABCDA_1B_1C_1D_1$, $M$ is the centre of the face $A_1B_1C_1D_1$ and $N$ is the centre of the face $BCC_1B_1$. Find the angle between the vectors $\overrightarrow{AM}$ and $\overrightarrow{AN}$.
In the regular tetrahedron $ABCD$, $AM_1$ is the median of the face $ABC$ and $AM_2$ is the median of the face $ACD$ ($M_1$ is the midpoint of $BC$, $M_2$ the midpoint of $CD$). Find the angle between $AM_1$ and $AM_2$.
Three forces $\vec F_1,\vec F_2,\vec F_3$, each of magnitude $a$, act at one point, and every two of them make an angle of $60^\circ$. Find the magnitude of the resultant and the angle it makes with each force.
Given the points $A(3,3,-2)$, $B(0,-3,4)$, $C(0,-3,0)$, $D(0,2,-4)$, find $\mathrm{Pr}_{\overrightarrow{CD}}\overrightarrow{AB}$, $\mathrm{Pr}_{\overrightarrow{AB}}\overrightarrow{CD}$ and the angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$.
The force $\vec F=5\vec i+4\vec j+3\vec k$ (in kg, i.e. kilogram-force) moves its point of application along the displacement $\vec S=2\vec i+\vec j-2\vec k$ (in m). Find the work done and the angle between $\vec F$ and $\vec S$.
A parallelogram is built on the vectors $\vec m=2\vec a-\vec b+\vec c$ and $\vec n=-3\vec a-\vec b+\vec c$, where $\vec a,\vec b,\vec c$ are unit vectors with $\angle(\vec a,\vec b)=\angle(\vec a,\vec c)=30^\circ$ and $\angle(\vec b,\vec c)=150^\circ$. Find the lengths of its diagonals.
Prove that the vector $\vec h$, which coincides with the height of the parallelogram constructed on the vectors $\vec a$ and $\vec b$ and is normal to $\vec a$, equals $\vec h=\vec b-(\vec a\cdot\vec b)\,\frac{\vec a_0}{|\vec a|},\qquad \vec a_0=\frac{\vec a}{|\vec a|}$.
Express the unit vector $\vec x$ through $\vec u$ and $\vec v$, given that $\vec x$ bisects the angle between $\vec a=\vec u+\vec v$ and $\vec b=2\vec u+\vec v$ and is coplanar with them, where $|\vec u|=2$, $|\vec v|=4$ and $\angle(\vec u,\vec v)=120^\circ$.
For the triangle $ABC$: $|\overrightarrow{AB}|=2$, $|\overrightarrow{AC}|=3$, $\angle(\overrightarrow{AB},\overrightarrow{AC})=\dfrac{\pi}{3}$. Determine the angle between the altitude and the median drawn from the vertex $A$.
A square piece of cardboard is divided into three equal strips and folded into a regular triangular prism. Find the angle between two adjacent segments into which the diagonal of the square is broken by the folding.
Given $\vec u=6\vec i+\vec j+\vec k$, $\vec v=3\vec j-\vec k$ and $\vec w=-2\vec i+3\vec j+5\vec k$, determine $\lambda$ so that $\vec u+\lambda\vec v$ and $\vec w$ are perpendicular.
A triangle is constructed on the vectors $\vec a=5\vec i+2\vec j$ and $\vec b=2\vec i-4\vec j$ ($\vec i,\vec j$ orthonormal). Find the altitude vector $\vec h_a$, drawn to the side $\vec a$, and the angle it makes with the side $\vec b$.
Given $\vec a\{0,2\lambda,\lambda\}$, $\vec b\{2,2,1\}$, $\vec c\{-1,-2,-1\}$ and $\vec d\{\lambda,0,1\}$. Determine $\lambda$ from the condition $\vec a\cdot\vec b=\vec c\cdot\vec d+7$.
Determine $\lambda$ so that the vector $\vec a=2\lambda\vec i+\vec j+(1-\lambda)\vec k$ makes equal angles with $\vec b=-\vec i+3\vec j$ and $\vec c=5\vec i-\vec j+8\vec k$.
In the plane $xOy$ find the vector $\vec r$ perpendicular to $\vec a\{5,-3,4\}$ with $|\vec r|=|\vec a|$. Can $\vec r$ be determined for an arbitrary vector $\vec a$?
Prove that for any points $A,B,C,D$ in space $\overrightarrow{BC}\cdot\overrightarrow{AD}+\overrightarrow{CA}\cdot\overrightarrow{BD}+\overrightarrow{AB}\cdot\overrightarrow{CD}=0$.
The position vectors of $A,B,C,D$ are $\vec r_1=\{5,2,-1\}$, $\vec r_2=\{1,-3,4\}$, $\vec r_3=\{-2,1,3\}$, $\vec r_4=\{2,6,-2\}$. $1^\circ$ Show that $ABCD$ is a parallelogram. $2^\circ$ Find the angle between its diagonals.
If $ABC$ and $PQR$ are two triangles in space, prove that their centroids coincide if and only if $\overrightarrow{AP}+\overrightarrow{BQ}+\overrightarrow{CR}=\vec 0$.
$ABCD$ and $A_0B_0C_0D_0$ are parallelograms, and the points $M,N,P,Q$ lie on the segments $AA_0,BB_0,CC_0,DD_0$ so that $\frac{AM}{AA_0}=\frac{BN}{BB_0}=\frac{CP}{CC_0}=\frac{DQ}{DD_0}=\lambda$. $1^\circ$ Show that $MNPQ$ is a parallelogram. $2^\circ$ What is the locus of the intersection of its diagonals as $\lambda$ varies?
Find the unit vector $\vec n_0$ of the vector $\vec n$ coplanar with $\vec p$ and $\vec q$, given $|\vec p|=2$, $|\vec q|=3$, $\angle(\vec p,\vec q)=\dfrac\pi3$, $\vec n\cdot\vec p=7$, $\vec n\cdot\vec q=3$.
A regular tetrahedron with edge $a$ lies in the first octant with one vertex at the origin $O$ and the edge $OA$ along $Ox$. $1^\circ$ Find the unit vectors along the edges $OA$, $OB$, $OC$. $2^\circ$ Forces of $1$, $2$ and $3\ \mathrm{kg}$ act along these edges. Find the magnitude of the resultant and the cosines of the angles it makes with each force.
Find the projection of $\vec m=\{\sqrt2,-3,-5\}$ onto an axis that makes $\alpha=45^\circ$ with $Ox$, $\gamma=60^\circ$ with $Oz$ and an obtuse angle $\beta$ with $Oy$.
Find $\vec c=\vec a\times\vec b$ and the area of the parallelogram constructed on $\vec a=2\vec i+3\vec j+0\vec k$ and $\vec b=0\vec i+3\vec j+2\vec k$.
Given $|\vec a|=5$, $|\vec b|=5$, $\angle(\vec a,\vec b)=\dfrac\pi4$, find the area of the parallelogram constructed on $2\vec b-\vec a$ and $3\vec a+2\vec b$.
Find the area of the parallelogram whose diagonals are $\vec d_1=3\vec m+3\vec n$ and $\vec d_2=\vec m-\vec n$, where $\vec m,\vec n$ are unit vectors with $\angle(\vec m,\vec n)=\dfrac\pi6$.
Can the signs $\times$ and $\cdot$ be interchanged in the scalar triple product, i.e. is $(\vec a\times\vec b)\cdot\vec c=\vec a\cdot(\vec b\times\vec c)$?
$\vec a=\{1,2\alpha,1\}$, $\vec b=\{2,\alpha,\alpha\}$, $\vec c=\{3\alpha,2,-\alpha\}$ are edges of a tetrahedron. $1^\circ$ Find its volume. $2^\circ$ Find $\alpha$ for which $\vec a,\vec b,\vec c$ are coplanar, and then decompose $\vec a$ along $\vec b$ and $\vec c$.
$A(2,0,0)$, $B(0,3,0)$, $C(0,0,6)$, $D(2,3,8)$ are the vertices of a pyramid. $1^\circ$ Compute its volume. $2^\circ$ Compute the height onto the base $ABC$.
$\vec a=\{\alpha,1,1\}$, $\vec b=\{1,\alpha,1\}$, $\vec c=\{1,1,\alpha\}$ are edges of a parallelepiped. $1^\circ$ Find its volume. $2^\circ$ Find the height vector $\vec h$ perpendicular to $\vec a$ and $\vec b$. $3^\circ$ For $\alpha\in[-2,1]$, find $\alpha$ giving the maximum and the minimum volume, those volumes, and the vectors when $V$ is minimal.
$\vec a=\{2\lambda,1,1-\lambda\}$, $\vec b=\{-1,3,0\}$, $\vec c=\{5,-1,8\}$. $1^\circ$ Find $\lambda$ so that $\vec a$ makes equal angles with $\vec b$ and $\vec c$. $2^\circ$ For that $\lambda$, find the inclination of $\vec c$ to the plane of $\vec a,\vec b$. $3^\circ$ Find the volume and one height of the parallelepiped on $\vec a,\vec b,\vec c$.
Compute the volume of the parallelepiped on $\vec m=\vec a+\vec b+\vec c$, $\vec n=\vec a+\vec b-\vec c$, $\vec p=\vec a-\vec b+\vec c$, if $\vec a,\vec b,\vec c$ form a right-handed triad and the parallelepiped on them has volume $\tfrac14$.
Find the unit vector normal to $\vec a=\{-2,-6,-1\}$ and $\vec b=\{1,2,0\}$ that makes an acute angle with $\vec c=\{2,-1,0\}$. In its direction find $\vec d$ such that $\vec a,\vec b,\vec d$ span a parallelepiped of volume $18$.
Triangle $ABC$ is given by $\overrightarrow{OA}=\vec a$, $\overrightarrow{OB}=\vec b$, $\overrightarrow{OC}=\vec c$. Find $\vec r=\overrightarrow{OM}$, where $M$ is an arbitrary point of the perpendicular bisector of $AB$.
If four points $A,B,C,D$ in space satisfy $\overline{AB}^2+\overline{CD}^2=\overline{AD}^2+\overline{BC}^2$, then $\overrightarrow{BD}\cdot\overrightarrow{AC}=0$. Prove this.
If four points $A,B,C,D$ in space satisfy $\overline{AB}^2+\overline{CD}^2=\overline{AD}^2+\overline{BC}^2$, then $\overrightarrow{BD}\cdot\overrightarrow{AC}=0$. Prove this.
Prove that any four vectors $\vec a,\vec b,\vec c,\vec d$ in space are linearly dependent: there exist $\alpha,\beta,\gamma,\delta$, not all zero, with $\alpha\vec a+\beta\vec b+\gamma\vec c+\delta\vec d=\vec 0$.
Decompose $\vec d=(3\vec a+\vec b-2\vec c)\times(\vec a-\vec b+5\vec c)$ along the mutually perpendicular unit vectors $\vec a,\vec b,\vec c$, which form a right-handed triad.
The opposite edges of a tetrahedron $ABCD$ are mutually perpendicular. Prove that $\cos(AB)\cos(CD)=\cos(AC)\cos(BD)=\cos(AD)\cos(BC)$ where $(XY)$ is the dihedral angle at the edge $XY$.
Let $\vec s_1,\dots,\vec s_4$ be the outward normals of the faces of a tetrahedron, with $|\vec s_i|$ equal to the area of the face. Prove that $\vec s_1+\vec s_2+\vec s_3+\vec s_4=\vec 0$.
Using vectors, derive $1^\circ\ \ \cos a=\cos b\cos c+\sin b\sin c\cos\alpha,\qquad 2^\circ\ \ \frac{\sin\alpha}{\sin a}=\frac{\sin\beta}{\sin b}=\frac{\sin\gamma}{\sin c}$ where $a,b,c$ are the sides of a spherical triangle and $\alpha,\beta,\gamma$ the opposite angles.
Prove that the area of the $n$-gon $A_1A_2\ldots A_n$ is $P=\tfrac12\Big|[\overrightarrow{OA_1}\times\overrightarrow{OA_2}]+[\overrightarrow{OA_2}\times\overrightarrow{OA_3}]+\cdots+[\overrightarrow{OA_n}\times\overrightarrow{OA_1}]\Big|$ where $O$ is any point in the plane and $[\ \cdot\times\cdot\ ]$ is the signed area of the parallelogram (positive for a counter-clockwise turn).
Is the following relation correct? $(\vec a\times\vec b)\times(\vec a\times\vec c)\stackrel{?}{=}(\vec a\cdot\vec b)\,[(\vec a\times\vec b)\cdot\vec c]$.
By translating the coordinate axes, reduce to canonical form and identify: $1^\circ$ $x^2+y^2-2x+4y+1=0$; $2^\circ$ $9x^2+4y^2+54x-8y+49=0$; $3^\circ$ $4x^2-16y^2+4x+16y-19=0$.
Find the affine transformation that maps the points $O(0,0)$, $A(1,0)$, $B(0,1)$ to $1^\circ$ $O'(1,-2)$, $A'(3,-1)$, $B'(0,1)$; $2^\circ$ $O'(0,0)$, $A'(1,-5)$, $B'(5,1)$.
Given the transformations $A_1:\ x'=3x-y,\ y'=x+y$ and $A_2:\ x'=-x+3y,\ y'=x+y$, find $A_1A_2$ and $A_2A_1$, and the invariant lines through the origin of $A_1$ and of $A_2$.
Find affine transformations that map the canonical curves $x'^2+y'^2=1$, $x'^2-y'^2=1$ and $y'^2=x'$ onto an ellipse, a hyperbola and a parabola with given centre (vertex), semi-axes (parameter) and axis direction $\theta$.
Find the affine transformation that maps the line $(a)\ x+y-1=0$ to $(a')\ x+y=0$, the line $(b)\ x-y=0$ to $(b')\ x-y-1=0$, and the point $A(0,0)$ to $A'(2,1)$.
Two coordinate systems of the same orientation are given: $xOy$ (rectangular) and $x'O'y'$ (oblique, coordinate angle $\omega=45^\circ$). The axis $O'x'$ makes an angle of $30^\circ$ with $Ox$, and $O'$ has coordinates $(1,-1)$ in the old system. Find the relations between the old and new coordinates.
By translation and rotation of the rectangular system $xOy$, reduce $2x^2-xy+2y^2-x+\lambda=0$ to canonical form and write the transformation formulas.
The axes of the new system $Ox'y'z'$ are the bisectors of the angles $\angle xOy$, $\angle yOz$, $\angle zOx$ of the rectangular system $Oxyz$. Find the transformation formulas (with the same unit of length on all axes).
The point $O'$ has coordinates $(1,2,-1)$ in the system $Oxyz$. The basis vectors of the system $O'x'y'z'$ are $\vec a\{-2,0,1\}$, $\vec b\{-1,-1,3\}$, $\vec c\{-1,-1,1\}$. Express the old coordinates through the new ones and find the coordinates of $O$ in the new system.
The old coordinates are expressed through the new ones by $3x=x'+2y'+2z'+3$, $3y=2x'+y'-2z'+6$, $3z=2x'-2y'+z'-3$. Find the equation of the surface $7x^2+6y^2+5z^2$ $-\,4yz-4xy$ $-\,6x-24y+18z+30=0$ in the new coordinates and identify it.
Find the affine transformation of space that maps: $1^\circ$ $M_1(0,0,0)$, $M_2(1,0,0)$, $M_3(0,1,0)$, $M_4(0,0,1)$ to $M_1'(3,5,-2)$, $M_2'(2,-1,-1)$, $M_3'(4,0,1)$, $M_4'(1,0,2)$; $2^\circ$ $M_1(3,5,-2)$, $M_2(-3,2,1)$, $M_3(0,3,0)$, $M_4(4,-2,0)$ to $M_1'(1,1,1)$, $M_2'(1,2,3)$, $M_3'(-1,-2,0)$, $M_4'(0,0,0)$.
Prove that under an affine transformation the ratio of segments on two parallel lines is preserved: if $\ell_1\parallel\ell_2$, $A,B\in\ell_1$, $C,D\in\ell_2$ ($C\ne D$) and $\overrightarrow{AB}=\lambda\,\overrightarrow{CD}$, then $\overrightarrow{A'B'}=\lambda\,\overrightarrow{C'D'}$.
Using the determinants $\delta$ and $\Delta$, determine the geometric meaning of: $1^\circ$ $x^2-4xy+3y^2-8x+14y+15=0$; $2^\circ$ $2x^2+3xy-5y^2=0$; $3^\circ$ $x^2-xy-y^2-x-y=0$.
For the family $\lambda x^2-3xy+y^2+\lambda x+3=0$, determine $\lambda$ so that the equation represents: $1^\circ$ ellipses (real or imaginary); $2^\circ$ hyperbolas; $3^\circ$ a parabola; $4^\circ$ a point; $5^\circ$ a pair of intersecting lines; $6^\circ$ a pair of parallel lines.
In the family $x^2-\alpha xy+y^2+4x+6y-\beta=0$, determine $\alpha$ and $\beta$ so that the curves pass through $A(2,0)$ and are parabolas. Sketch the parabolas.
For what values of $m$ and $n$ does $x^2+6xy+my^2+3x+ny-4=0$ represent: $1^\circ$ a central conic; $2^\circ$ a parabola; $3^\circ$ a pair of parallel lines?
In $x^2+\lambda xy+y^2-4x-4y+1=0$ determine $\lambda$ so that the equation represents: $1^\circ$ a pair of lines; $2^\circ$ a parabola; $3^\circ$ a rectangular hyperbola.
Write the equation of the plane through $M_1(2,-1,3)$ which: $1^\circ$ cuts equal intercepts on the coordinate axes; $2^\circ$ passes through the $x$-axis; $3^\circ$ passes through the origin and the point $(1,1,1)$.
Find the angle between the planes: $1^\circ$ $x+2z-6=0$ and $x+2y-4=0$; $2^\circ$ $\vec r\cdot(\vec i-2\vec j+2\vec k)=8$ and $\vec r\cdot(\vec i+\vec k)=6$.
Find the distance between the planes: $1^\circ$ $x-2y+z-1=0$ and $2x-4y+2z+1=0$; $2^\circ$ $\vec r\cdot(2\vec i+3\vec j-6\vec k)+14=0$ and $\vec r\cdot(2\vec i+3\vec j-6\vec k)-35=0$.
Through the intersection of $\pi_1:\ 4x-y+3z-1=0$ and $\pi_2:\ x+5y-z+2=0$, find the plane that: $1^\circ$ passes through $A(a,b,c)$; $2^\circ$ is parallel to the $y$-axis; $3^\circ$ is parallel to the $xOy$ plane; $4^\circ$ is perpendicular to $2x-y+5z-3=0$.
Write the equation of the plane through $M_1(2,5,-3)$ and the intersection of $\vec r\cdot(3\vec i-\vec j-\vec k)=5$ and $\vec r\cdot(\vec i+2\vec k)=0$.
What condition must hold among $A_i,B_i,C_i,D_i$ for the planes $A_ix+B_iy+C_iz+D_i=0$ ($i=1,2,3$) to belong to a pencil, i.e. to intersect along a single common line?
A line passes through $M_1(2,4,\lambda)$ and the point $(-3,0,0)$ of the plane $x+2y+13z+3=0$. Write its equation and determine $\lambda$ so that the line lies entirely in the plane.
Through $M_1(1,-2,1)$ draw a line: $1^\circ$ parallel to the $x$-axis; $2^\circ$ parallel to $\tfrac{x-2}{0}=\tfrac{y+2}{1}=\tfrac{z+3}{-3}$; $3^\circ$ parallel to $x-y+z-4=0,\ 2x+y-2z+5=0$; $4^\circ$ parallel to $x=2,\ y=3$.
Determine $n$ so that $\tfrac{x-1}{1}=\tfrac{y-2}{-1}=\tfrac{z-1}{n}$ intersects $\tfrac{x-2}{1}=\tfrac{y-3}{2}=\tfrac{z-4}{3}$, and find the intersection point.
A line has direction $\vec p$ and passes through $M_1$. Prove that the distance of $M_0$ from the line is $d=\frac{|\vec p\times\overrightarrow{M_1M_0}|}{|\vec p|} $.
Derive the formula for the shortest distance between the non-parallel lines $\frac{x-x_1}{l_1}=\frac{y-y_1}{m_1}=\frac{z-z_1}{n_1}\quad\text{and}\quad\frac{x-x_2}{l_2}=\frac{y-y_2}{m_2}=\frac{z-z_2}{n_2} $.
Through $M_1(1,2,-1)$ draw a line intersecting $\ell_1:\ \tfrac{x-1}{2}=\tfrac{y+1}{6}=\tfrac{z+3}{3}$ and $\ell_2:\ \tfrac{x-2}{3}=\tfrac y1=\tfrac{z+3}{-1}$.
A) Find the piercing point of $\tfrac{x-1}1=\tfrac y2=\tfrac{z+1}1$ through $x-2y+z+5=0$. B) Write the plane through $M_1(-1,2,-3)$ perpendicular to $\tfrac{x-1}1=\tfrac y2=\tfrac{z+2}1$, and find where the line pierces it.
What are the conditions for $1^\circ$ parallelism and $2^\circ$ perpendicularity of the line $\vec r\times\vec p=\vec q$ and the plane $\vec r\cdot\vec N=\alpha$?
For the line $\vec r\times\vec p=\vec q$ and the plane $\vec r\cdot\vec N=\alpha$, prove that the piercing point is $\vec r=\frac{\alpha\vec p+\vec N\times\vec q}{\vec p\cdot\vec N}$.
Write the plane through $\tfrac{x-3}2=\tfrac{y+4}1=\tfrac{z-2}{-3}$ parallel to $\tfrac{x+5}4=\tfrac{y-2}7=\tfrac{z-1}2$, and find the shortest distance between the lines.
Given $\ell_1:\ \tfrac{x-2}{\lambda}=\tfrac{y-1}1=\tfrac{z-2}0$ and $\ell_2:\ \tfrac{x-5}2=\tfrac{y-2}3=\tfrac{z-3}1$: $1^\circ$ determine $\lambda$ so that they intersect; $2^\circ$ find the intersection point; $3^\circ$ write the line through $O$ perpendicular to the plane of the two lines.
Given $\ell_1:\ x=a,\ y=b$ and $\ell_2:\ x+\lambda y+z=0,\ \lambda x-y-z=0$: $1^\circ$ find the common normal; $2^\circ$ find the locus of these normals as $\lambda$ varies.
Write the line that lies in the plane $x-4y+2z-7=0$, passes through the point where this plane meets the line $\ell$: $x-2y-4z+3=0$, $2x+y-3z+1=0$, and is perpendicular to $\ell$.
Write the line lying in $2x+3y-z+1=0$, passing through the point $M$ where the plane meets the $z$-axis, and making a minimal angle with $\tfrac{x-3}1=\tfrac{y-3}2=\tfrac z1$.
Write the line intersecting $\tfrac{x+5}2=\tfrac{y-3}{-4}=\tfrac{z+1}3$ and $\tfrac{x-3}{-2}=\tfrac{y+1}3=\tfrac{z+2}4$, and parallel to $3x+12y-3z-5=0$ and $3x-4y+9z+7=0$.
A point $M$ starts from $M_0(0,1,1)$ with velocity $\vec V=\{1,0,2\}$. Let $d_1$ and $d_2$ be its distances from the planes $\pi_1:\ x+4y-3z+9=0,\qquad \pi_2:\ 5x+z+8=0$ $1^\circ$ When is $d_1=d_2$? $2^\circ$ When is $d=d_1+d_2$ smallest?
A point $P$ moves so that $\overrightarrow{OP}=(1-t)\vec i+(2-3t)\vec j+(2t-1)\vec k$ $1^\circ$ Show that $P$ moves along a line $p$. $2^\circ$ Find the equation of $p$. $3^\circ$ When is $P$ in the plane $2x+3y+2z+1=0$? $4^\circ$ Find the plane normal to $p$ through the position at $t=2$.
Prove that the lines $p_1:\ \frac{x-l_1}{l}=\frac{y-m_1}{m}=\frac{z-n_1}{n},\qquad p_2:\ \frac{x-l}{l_1}=\frac{y-m}{m_1}=\frac{z-n}{n_1}$ $1^\circ$ always lie in one plane, and $2^\circ$ that this plane passes through the origin.
Find the line $p$ parallel to the plane $x+y=0$ that meets the lines below at two points $3$ apart: $p_1:\ \frac x3=\frac{y+1}1=\frac{z-3}{-2},\qquad p_2:\ y=z+2,\ x=1$.
$\vec r_2$ is the position vector of $P$. Find $1^\circ$ the distance from $P$ to the plane $(\vec r-\vec r_1)\cdot\vec a=0$, and $2^\circ$ the position vector $\vec r_0$ of the foot of the perpendicular from $P$.
Through the line $x+28y-2z+17=0,\ 5x+8y-z+1=0$, find the plane that $1^\circ$ is tangent to the sphere $x^2+y^2+z^2=1$, and $2^\circ$ cuts it in a great circle.
Find the planes tangent to $x^2+y^2+z^2-10x+2y+26z-113=0$ and parallel to the lines $\frac{x+5}2=\frac{y-1}{-3}=\frac{z+13}2,\qquad \frac{x+7}3=\frac{y+1}{-2}=\frac{z-8}0 $.
Find the cylindrical surface through the curve $(x-1)^2+(y+3)^2+(z-2)^2=25,\qquad x+y-z+2=0$ with generators parallel to $1^\circ$ the $x$-axis and $2^\circ$ the line $x=y,\ z=c$.
The curve $c$ is the intersection of the sphere $x^2+y^2+z^2=4(x-2y-2z)$ with the plane through its center normal to the line $x=0,\ y=-z$. Find the projection of $c$ onto $z=0$.
Given $c:\ y^2+z^2=1,\ x+z=0$: $1^\circ$ what curve is it? $2^\circ$ find its projections onto the coordinate planes; $3^\circ$ find the cone with vertex $S(1,1,1)$ and directrix $c$.
$A\in Ox$, $B\in Oy$, $OA=OB=a$. A line $p$ moves along the circle with diameter $AB$, meets $Oz$ and stays parallel to $z=0$. Find the conoid described by $p$.
The sphere $x^2+y^2+z^2=29$ rotates through the angle $\alpha$ about its diameter $MN$. The point $A(4,-3,2)$ moves to $A'(5,0,-2)$ along a great circle. Find: a) the line $MN$; b) the plane of rotation of $A$; c) $\alpha$; d) the angle between that plane and $z=0$; e) the projection of the great circle onto $z=0$.