Question

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=\langle 2,-3,1\rangle \\ \mathbf{v}=\langle-1,-1,-1\rangle \end{array} $$

Short answer

The vectors are orthogonal and not parallel.

Step-by-step solution

Check if the vectors are orthogonal

First, calculate the dot product of the vectors, \(\mathbf{u} \cdot \mathbf{v}\). Using the formula, \(\mathbf{u} \cdot \mathbf{v} = (2*-1) + (-3*-1) + (1*-1) = -2+3-1 = 0\). Since \(\mathbf{u} \cdot \mathbf{v} = 0\), this means that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Check if the vectors are parallel

We check if one vector is a scalar multiple of the other: \(\mathbf{v} = k\mathbf{u}\), \(\langle -1,-1,-1 \rangle = k\langle 2,-3,1 \rangle\). Solving this, we get different values for k for each component of the vector, so the vectors are not multiples of each other and hence not parallel.