Question

$\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors given $\overrightarrow{A}=2\hat{i}+3\hat{j}$ and $\overrightarrow{B}=\hat{i}+\hat{j}$ . The magnitude of the component $\overrightarrow{A}$ along $\overrightarrow{B}$ is :

Answer 1

The component of the vector $A$ along $B$ geometrically is given as,

$A\cos \theta =\left(\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{\left|B\right|}\right)\hat{B}$

$A\cos \theta =\left(\frac{\left(2\hat{i}+3\hat{j}\right)\cdot \left(\hat{i}+\hat{j}\right)}{\sqrt{1^2+1^2}}\right)\hat{B}$

$A\cos \theta =\left(\frac{5}{\sqrt{2}}\right)\hat{B}$

Thus, the magnitude of the component vector $A$ along $B$ is $\frac{5}{\sqrt{2}}$.