Question

Properties of Vectors

Let $u,v$ and $w$ be vectors, and let $c$ be a scalar. Prove the given property.

$(\mathbf{u}-\mathbf{v})·(\mathbf{u}+\mathbf{v})=|\mathbf{u}{|}^2-|\mathbf{v}{|}^2$

Short answer

The expression $(\mathbf{u}-\mathbf{v})·(\mathbf{u}+\mathbf{v})=|\mathbf{u}{|}^2-|\mathbf{v}{|}^2$ is an identity

Step-by-step solution

Step 1. Given information

Expression is given as-

$(\mathbf{u}-\mathbf{v})·(\mathbf{u}+\mathbf{v})=|\mathbf{u}{|}^2-|\mathbf{v}{|}^2$

Step 2. Concept used

The scalar product of two vectors is commutative.

Commutative Property:

$xy=yx$

Scalar product or Dot product:

$a·b=\left|a\right|\left|b\right|\cos \theta$

Simplify LHS and RHS separately of the expression. If both are equal then expression is an identity.

Step 3. Calculation

Let $u=⟨a,b⟩\&v=⟨p,q⟩$

Simplify LHS,

$\begin{array}{l}(u-v)(u+v)=u\cdot u+u\cdot v-v\cdot u-v\cdot v\\ (u-v)(u+v)=u\cdot u-v\cdot v\end{array}$

We know that,$a·a=|a{|}^2$

$(u-v)(u+v)=|u{|}^2-|v{|}^2$

Both LHS and RHS are equal. Hence, expression is an identity.