Question

Given the vectors A=4î + 8j and B= 5î + 10j, demostrate that the vectors are parallels.

Answer 1

When two vectors are parallel, angle between the vectors is zero.

$\bold{A}=4\hat{i}+8\hat{j}$   $\bold{B}=5\hat{i}+10\hat{j}$

$\bold{A}\cdot \bold{B}=AB\cos\theta$

$\cos\theta=\frac{\bold{A}\cdot \bold{B}}{AB}$

${\bold{A}\cdot \bold{B}}=(4\hat{i}+8\hat{j})\cdot(5\hat{i}+10\hat{j})=4*5+8*10=100$

$A=\sqrt{4^2+8^2}=\sqrt{80}$   $B=\sqrt{5^2+10^2}=\sqrt{125}$

$\theta=\cos^{-1}\left ( \frac{\bold{A}\cdot \bold{B}}{AB} \right )$

$\theta=\cos^{-1}\left ( \frac{100}{\sqrt{80}\sqrt{125}} \right )=\cos^{-1}(1)=0\degree$

Angle between the two vectors is $0\degree$ , so the given vectors are parallel.