First slide

Algebra of vectors

Question

$\vec{a}, \vec{b} and \vec{c} a r e t h r e e n o n z e r o n o n c o p l a n a r v e c t o r s a n d {\vec{r}}_{1} = p \vec{a} + q \vec{b} + \vec{c} and {\vec{r}}_{2} = \vec{a} + p \vec{b} + q \vec{c}$ . lf the vectors $\vec{r_{1}} + 2 \vec{r_{2}} and 2 {\vec{r}}_{1} + \vec{r_{2}}$ are collinear, then (p, q) is

Moderate

A

(0, 0)
B

(1, -1)
C

(-1, 1)
D

(1, 1)

Solution

$\begin{array}{l} \vec{r_{1}} + 2 {\vec{r}}_{2} = (p \vec{a} + q \vec{b} + \vec{c}) + 2 (\vec{a} + p \vec{b} + q \vec{c}) \\ = (p + 2) \vec{a} + (q + 2 p) \vec{b} + (1 + 2 q) \vec{c} \\ 2 \vec{r_{1}} + {\vec{r}}_{2} = (2 p + 1) \vec{a} + (2 q + p) \vec{b} + (2 + q) \vec{c} \\ \frac{p + 2}{2 p + 1} = \frac{q + 2 p}{2 q + p} = \frac{1 + 2 q}{2 + q} \\ = \frac{p + q + 2 p + 2 q + 3}{p + q + 2 p + 2 q + 3} = 1 \\ \Rightarrow p = 1 and q = 1 \end{array}$

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