Question

If $\overrightarrow{u}=3\hat{i}-5\hat{j}+9\hat{k}\text{and}\overrightarrow{v}=3\hat{i}+4\hat{j}+0k;$ What is the component of $\overrightarrow{u}$ along the direction of $\overrightarrow{v}$?

• A. $\frac{11}{7}$
• B. $\frac{-11}{5}$
• C. $\frac{7}{11}$
• D. $-5$

Answer 1

The correct option is B

$\frac{-11}{5}$

Component of $\overrightarrow{u}$ along the direction of $\overrightarrow{v}$ can be calculated using dot product.
We know,
$\overrightarrow{u}.\overrightarrow{v}=|\overrightarrow{u}||\overrightarrow{v}|cos\theta$
$\Rightarrow (9-20+0)=\left(\sqrt{(3{)}^2+(5{)}^2+(3{)}^2}\right)\left(\sqrt{(3{)}^2+(4{)}^2+(0{)}^2}\right).\cos \theta$
$\cos \theta =\frac{-11}{\sqrt{115}.5}$
The component of $\overrightarrow{u}$ along the direction of $\overrightarrow{v}$ will be = $|\overrightarrow{u}|$ $\cos \theta$
$|\overrightarrow{u}|=\sqrt{115}$
$\Rightarrow |\overrightarrow{u}|.\cos \theta =\frac{-11}{5}$