Question

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{aligned} &\mathbf{u}=\mathbf{j}+6 \mathbf{k}\\\ &\mathbf{v}=\mathbf{i}-2 \mathbf{j}-\mathbf{k} \end{aligned} $$

Short answer

The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are neither orthogonal nor parallel.

Step-by-step solution

Writing the vectors in component form

The vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be written as \(\mathbf{u} = <0,1,6>\) and \(\mathbf{v} = <1,-2,-1>\).

Calculating the dot product

Calculate the dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\). The dot product is calculated by multiplying the corresponding components together and then adding them up. The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is \(u \cdot v = (0*1) + (1*-2) + (6*-1) = -8\).

Determining whether the vectors are orthogonal, parallel or neither

If the dot product of two vectors is zero, the vectors are orthogonal. However, the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is not zero, so they're not orthogonal. Two vectors are parallel if one is a scalar multiple of the other. But neither of these vectors is a scalar multiple of the other, so they're not parallel. Therefore, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are neither orthogonal nor parallel.