Physics - Vectors Question with Solution | TestHub

Let the given vectors be $\vec{p}=a\hat{i}+b\hat{j}+\hat{k}$ and $\vec{q}=2\hat{i}-3\hat{j}+4\hat{k}$.

Since the vectors are perpendicular, their dot product is zero:

$\vec{p}\cdot \vec{q}=0$

$2a-3b+4=0$

$2a-3b=-4$

We are given another equation:

$3a+2b=7$

Solving the two equations simultaneously, multiply the first by $2$ and the second by $3$:

$4a-6b=-8$

$9a+6b=21$

Adding both equations:

$13a=13\Rightarrow a=1$

Substituting $a=1$ in $3a+2b=7$:

$3(1)+2b=7\Rightarrow 2b=4\Rightarrow b=2$

The ratio of $a$ to $b$ is $\frac{a}{b}=\frac{1}{2}$.

Given that the ratio is $\frac{x}{2}$, we have:

$\frac{x}{2}=\frac{1}{2}\Rightarrow x=1$

Answer: $1$