Question

Prove property$\left(5\right),\mathbf{0}\mathbf{·}\mathbf{v}=0.$

Short answer

It is proved that$\mathbf{0}\mathbf{·}\mathbf{v}=0.$

Step-by-step solution

Step 1. Given Information 

We have to prove the property that$\mathbf{0}\mathbf{·}\mathbf{v}=0.$

Step 2. Proving 

To prove the property let $\mathbf{v}=\mathrm{x}\mathbf{i}+\mathrm{y}\mathbf{j}$

Now, we know the dot product of two vectors is role="math" localid="1646825639826" $\mathbf{u}\mathbf{·}\mathit{w}={\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2$where $$\begin{gathered} \mathbf{u}={\mathrm{a}}_1\mathbf{i}+{\mathrm{b}}_1\mathbf{j} \\ \mathbf{w}\mathbf{=}{\mathrm{a}}_{\mathbf{2}}\mathbf{i}\mathbf{+}{\mathrm{b}}_{\mathbf{2}}\mathbf{j} \end{gathered}$$

So,

$$\begin{gathered} 0·\mathbf{v} \\ =0·\mathrm{x}+0·\mathrm{y} \\ =0 \end{gathered}$$

Hence proved.