Question

The Component of $u$ along $v$

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

$\mathbf{u}=7\mathbf{i},\mathbf{v}=8\mathbf{i}+6\mathbf{j}$

Short answer

The component is $5.6$

Step-by-step solution

Step 1. Given information

Vectors are given as-

$\mathbf{u}=7\mathbf{i},\mathbf{v}=8\mathbf{i}+6\mathbf{j}$

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 $v$,

$\begin{array}{l}{\mathrm{comp}}_{v}u=\frac{u\cdot v}{\left|v\right|}\\ =\frac{\left(7\right)\left(8\right)+\left(0\right)\left(6\right)}{\sqrt{8^2+6^2}}\\ =\frac{56}{10}\\ =5.6\end{array}$