Search any question & find its solution
Question:
Answered & Verified by Expert
$\mathbf{a}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\mathbf{b}$ are two vectors in $X O Y$ plane and $\mathbf{a}$ is perpendicular to $\mathbf{b}$. A vector $\mathbf{c}$ lying in the same plane and having projections 1 and 2 respectively $\mathbf{a}$ and $\mathbf{b}$ is
Options:
Solution:
1310 Upvotes
Verified Answer
The correct answer is:
$2 \hat{\mathbf{i}}-\hat{\mathbf{j}}$
Given,
$\begin{aligned}
\mathbf{a} & =4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}} \text { and } \mathbf{a} \cdot \mathbf{b}=0 \\
\frac{\mathbf{a} \cdot \mathbf{c}}{|\mathbf{a}|} & =1, \frac{\mathbf{b} \cdot \mathbf{c}}{|\mathbf{b}|}=2 \\
\mathbf{a} \cdot \mathbf{b} & =0 \\
\therefore \mathbf{b} & =3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}
\end{aligned}$
Let $\mathbf{c}=x \hat{\mathbf{i}}+\hat{y}$
$\begin{aligned}
& \therefore \frac{(4 \hat{\mathbf{i}}+3 \hat{\mathrm{j}}) \cdot(x \hat{\mathbf{i}}+\hat{y \hat{\mathrm{j}}})}{5}=1 \\
& =4 x+4 y=5
\end{aligned}$
and $\frac{\mathbf{b} \cdot \mathbf{c}}{|\mathrm{b}|}=2$
$\therefore 3 x-4 y=10$
On solving Eqs. (i) and (ii), we get
$\begin{aligned}
& x =2 y=-1 \\
\therefore & \mathbf{c}=2 \hat{\mathbf{i}}-\hat{\mathfrak{j}}
\end{aligned}$
$\begin{aligned}
\mathbf{a} & =4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}} \text { and } \mathbf{a} \cdot \mathbf{b}=0 \\
\frac{\mathbf{a} \cdot \mathbf{c}}{|\mathbf{a}|} & =1, \frac{\mathbf{b} \cdot \mathbf{c}}{|\mathbf{b}|}=2 \\
\mathbf{a} \cdot \mathbf{b} & =0 \\
\therefore \mathbf{b} & =3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}
\end{aligned}$
Let $\mathbf{c}=x \hat{\mathbf{i}}+\hat{y}$
$\begin{aligned}
& \therefore \frac{(4 \hat{\mathbf{i}}+3 \hat{\mathrm{j}}) \cdot(x \hat{\mathbf{i}}+\hat{y \hat{\mathrm{j}}})}{5}=1 \\
& =4 x+4 y=5
\end{aligned}$
and $\frac{\mathbf{b} \cdot \mathbf{c}}{|\mathrm{b}|}=2$
$\therefore 3 x-4 y=10$
On solving Eqs. (i) and (ii), we get
$\begin{aligned}
& x =2 y=-1 \\
\therefore & \mathbf{c}=2 \hat{\mathbf{i}}-\hat{\mathfrak{j}}
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.