Question

Suppose a vector \({\bf{y}}\) is orthogonal to vectors \({\bf{u}}\) and \({\bf{v}}\). Show that \({\rm{y}}\) is orthogonal to the vector \({\bf{u}} + {\bf{v}}\).

Short answer

It is verified that \({\bf{y}}\)is orthogonal to vector \({\bf{u}} + {\bf{v}}\).

Step-by-step solution

Definition of Orthogonal sets.

The two vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\) are Orthogonal if:

\(\begin{aligned}{l}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} = {\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2}\\{\rm{and}}\\{\bf{u}} \cdot {\bf{v}} = 0\end{aligned}\).

 Computing the required values.

Since, \({\bf{y}}\) is orthogonal to vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\).

Therefore,

\(\begin{aligned}{l}{\bf{y}} \cdot {\bf{u}} = 0\\{\bf{y}} \cdot {\bf{v}} = 0\end{aligned}\)

Using the inner productrule, we have:

\(\begin{aligned}{c}{\bf{y}} \cdot \left( {{\bf{u}} + {\bf{v}}} \right) &= {\bf{y}} \cdot {\bf{u}} + {\bf{y}} \cdot {\bf{v}}\\ &= 0 + 0\\ &= 0\end{aligned}\).

Hence proved: \({\bf{y}}\)is orthogonal to vector \({\bf{u}} + {\bf{v}}\).