Question

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=-2 \mathbf{i}+3 \mathbf{j}-\mathbf{k} \\ \mathbf{v}=2 \mathbf{i}+\mathbf{j}-\mathbf{k} \end{array} $$

Short answer

The vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal.

Step-by-step solution

Calculate the dot product

To check for orthogonality, calculate the dot product of \( \mathbf{u} \) and \( \mathbf{v} \). The formula for the dot product of \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) and \( \mathbf{v} = d\mathbf{i} + e\mathbf{j} + f\mathbf{k} \) is \( a*d + b*e + c*f \). In this case, \( \mathbf{u} \cdot \mathbf{v} = (-2*2) + (3*1) + (-1*-1) \).

Evaluate the dot product

Evaluating \( \mathbf{u} \cdot \mathbf{v} = (-2*2) + (3*1) + (-1*-1) \) gives us -4 + 3 + 1, which equals zero.

Check for parallelism

If the dot product is not zero, the next step would be to check if \( \mathbf{u} \) is a scalar multiple of \( \mathbf{v} \) to determine if they are parallel. But since the dot product is zero, this step is not necessary.