Write the trajectory of the point M that starts from…

Write the trajectory of the point $M$ that starts from $M_1(2,-3,-1)$ and moves rectilinearly with velocity $\vec u=\{2,3,1\}$.

Explanation of Method

The trajectory of a point that starts from M₁ and moves in a straight line with constant velocity.

Step 1: Solution

$\vec r(t)=\vec r_1+t\vec u$ for $t\ge0$: \[x=2+2t,\quad y=-3+3t,\quad z=-1+t\quad(t\ge0);\qquad \frac{x-2}{2}=\frac{y+3}{3}=\frac{z+1}{1}\]

The trajectory is the ray from $M_1$ in the direction $\vec u$. The speed is constant, $|\vec u|=\sqrt{14}$. At $t=1$ the point is at $(4,0,0)$, on the $x$-axis.

$\vec u$ $M_1\ (t=0)$

The trajectory is the ray from $M_1$ in the direction of $\vec u$ ($t\ge0$); after time $t=1$ the point is at $M_1+\vec u=(4,0,0)$.

Result

\[\boxed{\,x=2+2t,\quad y=-3+3t,\quad z=-1+t\quad(t\ge0);\qquad \frac{x-2}{2}=\frac{y+3}{3}=\frac{z+1}{1}\,}\]