Question

Determine which pairs of vectors in Exercises \(15 - 18\) are orthogonal.

16. \({\bf{u}} = \left( {\begin{aligned}{12}\\3\\{ - 5}\end{aligned}} \right),\,\,{\bf{v}} = \left( {\begin{aligned}{2}\\{ - 3}\\3\end{aligned}} \right)\)

Short answer

The vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\) are orthogonal to each other.

Step-by-step solution

Definition of orthogonal vectors

The two vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\) are Orthogonal if \({\bf{u}} \cdot {\bf{v}} = 0\).

Checking Orthogonality for given vectors.

The given vectors are:

\({\bf{u}} = \left( {\begin{aligned}{*{20}{r}}{12}\\3\\{ - 5}\end{aligned}} \right),\,\,{\bf{v}} = \left( {\begin{aligned}{*{20}{r}}2\\{ - 3}\\3\end{aligned}} \right)\)

On having dot products, we get:

\(\begin{aligned}{c}{\bf{u}} \cdot {\bf{v}} &= \left( {\begin{aligned}{*{20}{r}}{12}\\3\\{ - 5}\end{aligned}} \right).\left( {\begin{aligned}{*{20}{r}}2\\{ - 3}\\3\end{aligned}} \right)\\ &= \left( {12} \right)\left( 2 \right) + \left( 3 \right)\left( { - 3} \right) + \left( { - 5} \right)\left( 3 \right)\\ &= 24 - 9 - 15\\ &= 0\end{aligned}\)

Since \({\bf{u}} \cdot {\bf{v}} = 0\).

Hence, the vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\) are orthogonal to each other.