Vectors 5 Question 7
7. Let $\overrightarrow{\mathbf{b}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\overrightarrow{\mathbf{c}}$ be two vectors perpendicular to each other in the $X Y$-plane. All vectors in the same plane having projections 1 and 2 along $\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$, respectively are given by…. .
(1987, 2M)
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Answer:
Correct Answer: 7. $\pm \frac{1}{\sqrt{2}} \cdot(-\hat{\mathbf{j}}+\hat{\mathbf{k}})$
Solution:
- Let $\overrightarrow{\mathbf{c}}=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}$
Since, $\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are perpendiculars to each other. Then,
$$ \begin{aligned} & \quad \overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=0 \Rightarrow(4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}) \cdot(a \hat{\mathbf{i}}+b \hat{\mathbf{j}})=0 \\ & \Rightarrow \quad 4 a+3 b=0 \Rightarrow a: b=3:-4 \\ & \therefore \quad \overrightarrow{\mathbf{c}}=\lambda(3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}) \text {, where } \lambda \text { is constant of ratio. } \end{aligned} $$
Let the required vectors be $\overrightarrow{\mathbf{a}}=p \hat{\mathbf{i}}+q \hat{\mathbf{j}}$
Projection of $\overrightarrow{\mathbf{a}}$ on $\overrightarrow{\mathbf{b}}$ is $\frac{\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}}{|\overrightarrow{\mathbf{b}}|}$
$\therefore \quad 1=\frac{4 p+3 q}{5} \Rightarrow 4 p+3 q=5$
Also, projection of $\overrightarrow{\mathbf{a}}$ on $\overrightarrow{\mathbf{c}}$ is $\frac{\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}}}{|\overrightarrow{\mathbf{c}}|}$
$$ \Rightarrow \quad 2=\frac{3 \lambda p-4 \lambda q}{5 \lambda} \Rightarrow 3 p-4 q=10 $$
On solving above equations, we get $p=2, q=-1$
$$ \therefore \quad \overrightarrow{\mathbf{c}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}} $$
