Question

The non-zero vectors $\overline{a},\overline{b}$ and $\overline{c}$ are related $\overline{a}=8\overline{b}$ and $\overline{c}=-7\overline{b}$. Then the angle between $\overline{a}$ and $\overline{c}$

• A. $0$
• B. $\pi /4$
• C. $\pi /2$
• D. $\pi$

Answer 1

The correct option is D $\pi$

Consider the problem

Given that,

$\overrightarrow{a}=8\overrightarrow{b}$ and $\overrightarrow{c}=-7\overrightarrow{b}$

Let

$b=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$

Then,

$\overrightarrow{a}=8(b_1\hat{i}+b_2\hat{j}+b_3\hat{k})$

And

$\overrightarrow{c}=-7(b_1\hat{i}+b_2\hat{j}+b_3\hat{k})$

Consider that angle between $\overrightarrow{a}$ and $\overrightarrow{c}$ is $x$.

Thus,

$\cos x=\frac{a\cdot c}{\left|a\right|\left|c\right|}$

$\cos x=\frac{-56\left(b_1^2+b_2^2+b_3^2\right)}{7\times 8\sqrt{b_1^2+b_2^2+b_3^2}.\sqrt{b_1^2+b_2^2+b_3^2}}$

$\cos x=-1$

Therefore,

$x=\pi$

hence angle between $\overrightarrow{a}$ and $\overrightarrow{c}$ is $\pi$