91Ó°ÊÓ

Find the distance from the point \((-2,1,4)\) to the \begin{equation} \text{a.plane x=3} \quad \text { b. plane } y=-5 \quad \text { c. plane } z=-1 \end{equation}

Find a plane through the points \(P_{1}(1,2,3), P_{2}(3,2,1)\) and perpendicular to the plane \(4 x-y+2 z=7.\)

Sketch the surfaces in Exercises \(13-44\) . HYPERBOLIC PARABOLOIDS $$ x^{2}-y^{2}=z $$

Cross products of three vectors Show that except in degenerate cases, \(( \mathbf { u } \times \mathbf { v } ) \times \mathbf { w }\) lies in the plane of \(\mathbf { u }\) and \(\mathbf { v } ,\) whereas \(\mathbf { u } \times ( \mathbf { v } \times \mathbf { w } )\) lies in the plane of \(\mathbf { v }\) and \(\mathbf { w } .\) What are the degenerate cases?

Find the distance from the point \((4,3,0)\) to the \begin{equation} \text { a. }x \text { -axis } \quad \text { b. } y \text { -axis } \quad \text { c. } z \end{equation}

Cancelation in cross products If \(\mathbf { u } \times \mathbf { v } = \mathbf { u } \times \mathbf { w }\) and \(\mathbf { u } \neq \mathbf { 0 }\) then does \(\mathbf { v } = \mathbf { w } ?\) Give reasons for your answer.

find the distance from the point to the line. \begin{equation}(0,0,12) ; \quad x=4 t, \quad y=-2 t, \quad z=2 t\end{equation}

Line perpendicular to a vector Show that \(v=a \mathbf{i}+b \mathbf{j}\) is perpendicular to the line \(a x+b y=c\) by establishing that the slope of the vector \(v\) is the negative reciprocal of the slope of the given line.

Sketch the surfaces in Exercises \(13-44\) . ASSORTED $$ z=1+y^{2}-x^{2} $$

Find a vector of magnitude 7 in the direction of \(\mathbf{v}=12 \mathbf{i}-5 \mathbf{k}\)