Question

Prove the distributive property:

$\mathbf{u}\mathbf{·}\left(v+w\right)\mathbf{=}\mathbf{u}\mathbf{·}\mathbf{v}\mathbf{+}\mathbf{u}\mathbf{·}\mathbf{w}$

Short answer

It is proved that$\mathbf{u}\mathbf{·}\mathbf{(}\mathbf{v}\mathbf{+}\mathbf{w}\mathbf{)}\mathbf{=}\mathbf{u}\mathbf{·}\mathbf{v}\mathbf{+}\mathbf{u}\mathbf{·}\mathbf{w}$.

Step-by-step solution

Step 1. Given Information

We have to prove the distributive property$\mathbf{u}\mathbf{·}\mathbf{(}\mathbf{v}\mathbf{+}\mathbf{w}\mathbf{)}\mathbf{=}\mathbf{u}\mathbf{·}\mathbf{v}\mathbf{+}\mathbf{u}\mathbf{·}\mathbf{w}\mathbf{.}$

Step 2. Proving

To prove the distributive property, let

$$\begin{gathered} \mathbf{u}=\mathrm{a}\mathbf{i}\mathbf{}+\mathrm{b}\mathbf{j} \\ \mathbf{v}=\mathrm{x}\mathbf{i}\mathbf{}+\mathrm{y}\mathbf{j} \\ \mathbf{w}=\mathrm{m}\mathbf{i}\mathbf{}+\mathrm{n}\mathbf{j} \end{gathered}$$

Now, take L.H.S

role="math" localid="1646825259317" $$\begin{gathered} \mathbf{u}\mathbf{·}\mathbf{(}\mathbf{v}\mathbf{+}\mathbf{w}\mathbf{)} \\ =\left(a\mathbf{i}+b\mathbf{j}\right)·\left(x\mathit{i}+y\mathbf{j}+m\mathbf{i}+n\mathbf{j}\right) \\ =\left(a\mathbf{i}+b\mathbf{j}\right)·\left(\left(x+m\right)\mathbf{i}+\left(y+\mathrm{n}\right)\mathbf{j}\right) \\ =a\left(x+m\right)+b\left(y+n\right) \\ =ax+am+by+bn \\ =\left(ax+by\right)+\left(am+bn\right) \end{gathered}$$

R.H.S

$$\begin{gathered} \mathbf{u}\mathbf{·}\mathbf{v}\mathbf{}\mathbf{+}\mathbf{}\mathbf{u}\mathbf{·}\mathbf{w} \\ =\left(\mathrm{a}\mathbf{i}+\mathrm{b}\mathbf{j}\right)·\left(\mathrm{x}\mathbf{i}+\mathrm{y}\mathbf{j}\right)+\left(\mathrm{a}\mathbf{i}+\mathrm{b}\mathbf{j}\right)·\left(\mathrm{m}\mathbf{i}+\mathrm{n}\mathbf{j}\right) \\ =\left(\mathrm{ax}+\mathrm{by}\right)+\left(\mathrm{am}+\mathrm{bn}\right) \end{gathered}$$

L.H.S = R.H.S