Given vectors p=2i-3j , q=-i+2j and a=9i+4j , where i and j…

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 $\vec{q}$.

Step 1: Statement

We seek scalars $\lambda$, $\mu$ such that $\vec{a} = \lambda\vec{p} + \mu\vec{q}$, i.e. $\vec a$ is the sum of a vector along $\vec p$ and a vector along $\vec q$.

Step 2: Set up the system

\[ 9\vec{i}+4\vec{j} = \lambda(2\vec{i}-3\vec{j}) + \mu(-\vec{i}+2\vec{j}) = (2\lambda-\mu)\vec{i} + (-3\lambda+2\mu)\vec{j} \]

\[ \begin{cases} 2\lambda - \mu = 9 & (i)\\ -3\lambda + 2\mu = 4 & (ii) \end{cases} \]

Step 3: Solve

From $(i)$: $\mu = 2\lambda - 9$. Substituting into $(ii)$: \[-3\lambda + 4\lambda - 18 = 4 \;\Longrightarrow\; \lambda = 22, \qquad \mu = 44 - 9 = 35.\]

\[\vec{a} = 22\,\vec{p} + 35\,\vec{q}\]

The components along the two directions are

\[ \vec{a}_p = 22(2\vec{i}-3\vec{j}) = 44\vec{i}-66\vec{j}, \qquad \vec{a}_q = 35(-\vec{i}+2\vec{j}) = -35\vec{i}+70\vec{j}. \]

Step 4: Verification

\[ \vec{a}_p + \vec{a}_q = (44-35)\vec{i}+(-66+70)\vec{j} = 9\vec{i}+4\vec{j} = \vec{a} \;\checkmark \]

Step 5: Magnitudes

\[ |\vec{a}_p| = 22\sqrt{13}\approx 79.3, \qquad |\vec{a}_q| = 35\sqrt{5}\approx 78.3, \qquad |\vec{a}| = \sqrt{97}\approx 9.85 \]

Step 6: Angle between $\vec{p}$ and $\vec{q}$

\[ \cos\theta = \frac{\vec{p}\cdot\vec{q}}{|\vec{p}||\vec{q}|} = \frac{(2)(-1)+(-3)(2)}{\sqrt{13}\,\sqrt{5}} = -\frac{8}{\sqrt{65}} \approx -0.992, \qquad \theta \approx 172.9^\circ \]

$\vec p$ and $\vec q$ are almost antiparallel. That is why the coefficients are so large: two long, nearly opposite components almost cancel, leaving the short vector $\vec a$. The decomposition is oblique (not orthogonal).

Step 7: Graphical construction