JEE MAIN - Physics (2023 - 24th January Morning Shift - No. 21)
Vectors $$a\widehat i + b\widehat j + \widehat k$$ and $$2\widehat i - 3\widehat j + 4\widehat k$$ are perpendicular to each other when $$3a + 2b = 7$$, the ratio of $$a$$ to $$b$$ is $${x \over 2}$$. The value of $$x$$ is ____________.
Answer
1
Explanation
For two perpendicular vectors
$$ \begin{aligned} & (a \hat{i}+b \hat{j}+\hat{k}) \cdot(2 \hat{i}-3 \hat{j}+4 \hat{k})=0 \\\\ & 2 a-3 b+4=0 \end{aligned} $$
On solving, $2 a-3 b=-4$
Also given
$$ 3 a+2 b=7 $$
We get $\mathrm{a}=1, \mathrm{~b}=2$
$$ \begin{aligned} & \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{x}}{2} \Rightarrow \mathrm{x}=\frac{2 \mathrm{a}}{\mathrm{b}}=\frac{2 \times 1}{2} \\\\ & \Rightarrow \mathrm{x} =1 \end{aligned} $$
$$ \begin{aligned} & (a \hat{i}+b \hat{j}+\hat{k}) \cdot(2 \hat{i}-3 \hat{j}+4 \hat{k})=0 \\\\ & 2 a-3 b+4=0 \end{aligned} $$
On solving, $2 a-3 b=-4$
Also given
$$ 3 a+2 b=7 $$
We get $\mathrm{a}=1, \mathrm{~b}=2$
$$ \begin{aligned} & \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{x}}{2} \Rightarrow \mathrm{x}=\frac{2 \mathrm{a}}{\mathrm{b}}=\frac{2 \times 1}{2} \\\\ & \Rightarrow \mathrm{x} =1 \end{aligned} $$
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