Question

The Component of $u$ along $v$

Find the component of $u$ along $v$.

$\mathbf{u}=⟨-3,5⟩,\mathbf{v}=⟨1/\sqrt{2},1/\sqrt{2}⟩$

Short answer

The component is $\sqrt{2}$

Step-by-step solution

Step 1. Given information

Vectors are given as-

$\mathbf{u}=⟨-3,5⟩,v=⟨1/\sqrt{2},1/\sqrt{2}⟩$

Step 2. Concept used

The dot product of two vectors $u$ and $v$ can be defined as-

$u·v=a_1{b}_1+a_2{b}_2$

Where $u=\left(a_1,a_2\right)\&v=\left(b_1,b_2\right)$

If $u$ and $v$ are two vectors then the component of $u$ along $v$ can be defined as $com{p}_{v}u=\frac{u·v}{\left|v\right|}$

Use the formula of components of $u$ along $v$ and the dot product of vectors.

Step 3. Calculation

Component of $u$ along $v$,

$\begin{array}{l}{\mathrm{comp}}_{v}u=\frac{u\cdot v}{\left|v\right|}\\ =\frac{-3\left\{\frac{1}{\sqrt{2}}\right\}+5\left\{\frac{1}{\sqrt{2}}\right\}}{\sqrt{{\left\{\frac{1}{\sqrt{2}}\right\}}^2+{\left\{\frac{1}{\sqrt{2}}\right\}}^2}}\\ =\frac{\sqrt{2}}{1}\\ =\sqrt{2}\end{array}$