Question

Dot Products

Find the indicated quantity, assuming that $\mathbf{u}=2\mathbf{i}+\mathbf{j},\mathbf{v}=\mathbf{i}-3\mathbf{j}\&\mathbf{w}=3\mathbf{i}+4\mathbf{j}$

$(\mathbf{u}+\mathbf{v})·(\mathbf{u}-\mathbf{v})$

Short answer

The required value of dot product is $-5$

Step-by-step solution

Step 1. Given information

Vectors are given as

$\begin{array}{l}u=2i+j\\ v=i-3j\\ w=3i+4j\end{array}$

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

Properties of the dot product of vectors

$u·u=|u{|}^2$

Transform the required expression and find the value using the dot product of vectors.

Step 3. Calculation

Using Properties of the dot product of vectors,

$\begin{array}{l}(u+v)\cdot (u-v)=(u\cdot u)-(v\cdot v)\\ =|u{|}^2-|v{|}^2\\ ={\left\{\sqrt{2^2+1^2}\right\}}^2-{\left\{\sqrt{1^2+{(-3)}^2}\right\}}^2\\ =5-10\\ =-5\end{array}$