Question 26: Motion in 2D & Vectors

Question

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 ____________.

Accepted Answer

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 $\mathrm{a}=1, \mathrm{~b}=2$
$$
\begin{aligned}
& \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{x}}{2} \Rightarrow \mathrm{x}=\frac{2 \mathrm{a}}{\mathrm{b}}=\frac{2 	imes 1}{2} \\
& \Rightarrow \mathrm{x} =1
\end{aligned}
$$

Motion in 2D & Vectors

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 ____________.

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