Question

The Component of $u$ along $v$

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

$\mathbf{u}=⟨4,6⟩,\mathbf{v}=⟨3,-4⟩$

Short answer

The component is $-\frac{12}{5}$

Step-by-step solution

Step 1. Given information

Vectors are given as

$\mathbf{u}=⟨4,6⟩,\mathbf{v}=⟨3,-4⟩$

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 $v$,

$\begin{array}{l}{\mathrm{comp}}_{v}u=\frac{u\cdot v}{\left|v\right|}\\ =\frac{\left(4\right)\left(3\right)+\left(6\right)(-4)}{\sqrt{3^2+{(-4)}^2}}\\ =\frac{12-24}{5}\\ =\frac{-12}{5}\end{array}$