Physics - Vectors Question with Solution | TestHub

Given vectors are $\vec{p}=a\hat{i}+b\hat{j}+\hat{k}$ and $\vec{q}=2\hat{i}-3\hat{j}+4\hat{k}$.

Since the vectors are perpendicular to each other, their dot product is zero.

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

$2a-3b+4=0$

$2a-3b=-4$

We are also given the equation:

$3a+2b=7$

Multiplying the first equation by $2$ and the second equation by $3$, we get:

$4a-6b=-8$

$9a+6b=21$

Adding these two equations gives:

$13a=13\Rightarrow a=1$

Substituting $a=1$ into $3a+2b=7$ gives:

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

The ratio of $a$ to $b$ is:

$\frac{a}{b}=\frac{1}{2}$

It is given that the ratio is $\frac{x}{2}$.

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

Answer: 1