Vectors — Maths STD 12 Science — Question

Since, $\bar{c}$ is parallel to $\bar{b}, \bar{c}=m \bar{b}$, where $m$ is a scalar.

$\therefore \bar{c}=m(3 \hat{i}+\hat{k})$

i.e. $\bar{c}=3 m \hat{i}+m \hat{k}$

Let $\bar{d}=x \hat{i}+y \hat{j}+z \hat{k}$

Since, $\bar{d}$ is perpendicular to $\bar{b}=3 \hat{i}+\hat{k}, \bar{d} \cdot \bar{b}=0$

$\begin{aligned} & \therefore(x \hat{i}+y \hat{j}+z \hat{k}) \cdot(3 \hat{i}+\hat{k})=0 \\ & \therefore 3 x+z=0 \quad \therefore z=-3 x \\ & \therefore \bar{d}=x \hat{i}+y \hat{k}-3 x \hat{k}\end{aligned}$

Now, $\bar{a}=\bar{c}+\bar{d}$ gives

$\begin{aligned} \therefore 5 \hat{i}-2 \hat{j}+5 \hat{k} & =(3 m \hat{i}+m \hat{k})+(x \hat{i}+y \hat{j}-3 x \hat{k}) \\ & =(3 m+x) \hat{i}+y \hat{j}+(m-3 x) \hat{k}\end{aligned}$

By equality of vectors 3m + x = 5 … (1) y = -2 and m – 3x = 5 From (1) and (2) 3m + x = m – 3x ∴ 2m = -4x m ∴ m = -2x Substituting m = -2x in (1), we get ∴ -6x + x = 5 ∴ -5x = 5 ∴ x = -1 ∴ m = -2x = 2

$\therefore \bar{c}=6 \hat{i}+2 \hat{k}$ is parallel to $\hat{b}$ and

$\bar{d}=-\hat{i}-2 \hat{j}+3 \hat{k}$ is perpendicular to $\bar{b}$

Hence, $\bar{a}=\bar{c}+\bar{d}$, where $\bar{c}=6 \hat{i}+2 \hat{k}$ and $\bar{d}=-\hat{i}-2 \hat{j}+3 \hat{k}$.