Question

Properties of Vectors

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

$\mathbf{u}·\mathbf{v}=\mathbf{v}·\mathbf{u}$

Short answer

The expression $\mathbf{u}·\mathbf{v}=\mathbf{v}·\mathbf{u}$is an identity

Step-by-step solution

Step 1. Given information

Expression is given as-

$\mathbf{u}·\mathbf{v}=\mathbf{v}·\mathbf{u}$

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$u=⟨a,b⟩\&v=⟨c,d⟩$

Simplify LHS,

$\begin{array}{l}u\cdot v=⟨a,b⟩⟨c,d⟩\\ =ac+bd\end{array}$

Simplify RHS,

$\begin{array}{l}v\cdot u=⟨c,d⟩⟨a,b⟩\\ =ca+db\end{array}$

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