91Ó°ÊÓ

In Exercises 9 and \(10,\) write the equation of the plane passing through \(P\) with direction vectors \(\mathbf{u}\) and \(\mathbf{v}\) in \((a)\) vector form and (b) parametric form. $$P=(4,-1,3), \mathbf{u}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} -1 \\ 1 \\ 1 \end{array}\right]$$

Find the equation of the set of all points that are equidistant from the points \(P=(1,0,-2)\) and \(Q=(5,2,4)\)

(a) Prove that if \(\mathbf{u}\) is orthogonal to both \(\mathbf{v}\) and \(\mathbf{w}\), then \(\mathbf{u}\) is orthogonal to \(\mathbf{v}+\mathbf{w}\) (b) Prove that if \(\mathbf{u}\) is orthogonal to both \(\mathbf{v}\) and \(\mathbf{w}\), then \(\mathbf{u}\) is orthogonal to \(s \mathbf{v}+t \mathbf{w}\) for all scalars \(s\) and \(t\)