Question

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=\langle\cos \theta, \sin \theta,-1\rangle \\\\\mathbf{v}=\langle\sin \theta,-\cos \theta, 0\rangle \end{array} $$

Short answer

The vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal, but not parallel.

Step-by-step solution

Calculate the Dot Product

The dot product of two vectors \( \mathbf{u}=\langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v}=\langle v_1, v_2, v_3 \rangle \) is defined as \( u_1 * v_1 + u_2 * v_2 + u_3 * v_3 \). Using this formula, let's find the dot product of \( \mathbf{u} \) and \( \mathbf{v} \). It is \(\cos \theta * \sin \theta + \sin \theta * -\cos \theta + -1 * 0 = 0\).

Check for Orthogonality

If the dot product of two vectors is zero, the vectors are orthogonal. Since the dot product of \( \mathbf{u} \) and \( \mathbf{v} \) is zero, the vectors are orthogonal.

Check for Parallelism

If the vectors \( \mathbf{u} \) and \( \mathbf{v} \) were parallel, \( \mathbf{v} \) would be a scalar multiple of \( \mathbf{u} \). However, no scalar \( k \) exists such that \( k \) * \( \mathbf{u} = \mathbf{v} \). Hence, the vectors are not parallel.