Question

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

17. \({\bf{u}} = \left( {\begin{aligned}{3}\\2\\{ - 5}\\0\end{aligned}} \right),\,\,{\bf{v}} = \left( {\begin{aligned}{ - 4}\\1\\{ - 2}\\6\end{aligned}} \right)\)

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:

\({\bf{u}} = \left( {\begin{aligned}{*{20}{r}}3\\2\\{ - 5}\\0\end{aligned}} \right),\,\,{\bf{v}} = \left( {\begin{aligned}{*{20}{r}}{ - 4}\\1\\{ - 2}\\6\end{aligned}} \right)\)

On having dot products, we get:

\(\begin{aligned}{c}{\bf{u}} \cdot {\bf{v}} = \left( {\begin{aligned}{*{20}{r}}3\\2\\{ - 5}\\0\end{aligned}} \right) \cdot \left( {\begin{aligned}{*{20}{r}}{ - 4}\\1\\{ - 2}\\6\end{aligned}} \right)\\ = \left( 3 \right)\left( { - 4} \right) + \left( 2 \right)\left( 1 \right) + \left( { - 5} \right)\left( { - 2} \right) + \left( 0 \right)\left( 6 \right)\\ = - 12 + 2 + 10 + 0\\ = 0\end{aligned}\)

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

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