Question

Verify the parallelogram law for vectors \({\bf{u}}\) and \({\bf{v}}\) in \({\mathbb{R}^3}\):\({\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} + {\left\| {{\bf{u}} - {\bf{v}}} \right\|^2} = 2{\left\| {\bf{u}} \right\|^2} + 2{\left\| {\bf{v}} \right\|^2}\).

Short answer

It is verified that, \({\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} + {\left\| {{\bf{u}} - {\bf{v}}} \right\|^2} = 2{\left\| {\bf{u}} \right\|^2} + 2{\left\| {\bf{v}} \right\|^2}\).

Step-by-step solution

 Find \({\left\| {\bf{u}} \right\|^2},{\left\| {\bf{v}} \right\|^2},{\rm{ and }}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2},{\left\| {{\bf{u}} - {\bf{v}}} \right\|^2}\)

Find \({\left\| {\bf{u}} \right\|^2}\) and \({\left\| {\bf{v}} \right\|^2}\).

\(\begin{array}{l}{\left\| {\bf{u}} \right\|^2} &= {\bf{u}} \cdot {\bf{u}}\\{\left\| {\bf{v}} \right\|^2} &= {\bf{v}} \cdot {\bf{v}}\end{array}\)

Find \({\left\| {{\bf{u}} + {\bf{v}}} \right\|^2}\) and \({\left\| {{\bf{u}} - {\bf{v}}} \right\|^2}\).

\(\begin{aligned}{c}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} &= \left( {{\bf{u}} + {\bf{v}}} \right) \cdot \left( {{\bf{u}} + {\bf{v}}} \right)\\ &= {\bf{u}} \cdot {\bf{u}} + 2{\bf{u}} \cdot {\bf{v}} + {\bf{v}} \cdot {\bf{v}}\end{aligned}\)

\(\begin{aligned}{c}{\left\| {{\bf{u}} - {\bf{v}}} \right\|^2} &= \left( {{\bf{u}} - {\bf{v}}} \right) \cdot \left( {{\bf{u}} - {\bf{v}}} \right)\\ &= {\bf{u}} \cdot {\bf{u}} - 2{\bf{u}} \cdot {\bf{v}} + {\bf{v}} \cdot {\bf{v}}\end{aligned}\)

 Find the sum of obtained values

Find the sum of \({\left\| {\bf{u}} \right\|^2}\) and \({\left\| {\bf{v}} \right\|^2}\).

\({\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2} = {\bf{u}} \cdot {\bf{u}} + {\bf{v}} \cdot {\bf{v}}{\rm{ }}\left( 1 \right)\)

Find the sum of \({\left\| {{\bf{u}} + {\bf{v}}} \right\|^2}\) and \({\left\| {{\bf{u}} - {\bf{v}}} \right\|^2}\).

\(\begin{aligned}{c}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} + {\left\| {{\bf{u}} - {\bf{v}}} \right\|^2} &= \left( {{\bf{u}} + {\bf{v}}} \right) \cdot \left( {{\bf{u}} + {\bf{v}}} \right) + \left( {{\bf{u}} - {\bf{v}}} \right) \cdot \left( {{\bf{u}} - {\bf{v}}} \right)\\ &= {\bf{u}} \cdot {\bf{u}} + 2{\bf{u}} \cdot {\bf{v}} + {\bf{v}} \cdot {\bf{v}} + {\bf{u}} \cdot {\bf{u}} - 2{\bf{u}} \cdot {\bf{v}} + {\bf{v}} \cdot {\bf{v}}\\ &= 2{\bf{u}} \cdot {\bf{u}} + 2{\bf{v}} \cdot {\bf{v}}\end{aligned}\)

Use equation (1) in the obtained expression.

\({\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} + {\left\| {{\bf{u}} - {\bf{v}}} \right\|^2} = 2{\left\| {\bf{u}} \right\|^2} + 2{\left\| {\bf{v}} \right\|^2}\)

Hence, the parallelogram law is proved.