First slide

Algebra of vectors

Question

Given three vectors $\vec{a} = 6 \hat{i} - 3 \hat{j}, \vec{b} = 2 \hat{i} - 6 \hat{j} and \vec{c} = - 2 \hat{i} + 21 \hat{j}$ such that $\vec{α} = \vec{a} + \vec{b} + \vec{c}$ resolution of the vector $\vec{α}$ components with respect to $\vec{a} and \vec{b}$ is given by

Moderate

A

$3 \vec{a} - 2 \vec{b}$
B

$3 \vec{b} - 2 \vec{a}$
C

$2 \vec{a} - 3 \vec{b}$
D

$\vec{a} - 2 \vec{b}$

Solution

$\begin{array}{l} \vec{α} = \vec{a} + \vec{b} + \vec{c} = 6 \hat{i} + 12 \hat{j} \\ Let \vec{α} = x \vec{a} + y \vec{b} \Rightarrow 6 x + 2 y = 6 \\ and - 3 x - 6 y = 12 \\ ∴ x = 2, y = - 3 \\ ∴ \vec{α} = 2 \vec{a} - 3 \vec{b} \end{array}$

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