91Ó°ÊÓ

Determine whether the statement is true or false. Explain your answer. Given a nonzero scalar \(c\) and vectors \(\mathbf{b}\) and \(\mathbf{d},\) the vector equation \(c \mathbf{a}+\mathbf{b}=\mathbf{d}\) has a unique solution a.

Show that two nonzero vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) are orthogonal if and only if their direction cosines satisfy \(\quad \cos \alpha_{1} \cos \alpha_{2}+\cos \beta_{1} \cos \beta_{2}+\cos \gamma_{1} \cos \gamma_{2}=0\)

An equation is given in cylindrical coordinates. Express the equation in rectangular coordinates and sketch the graph. $$ \theta=\pi / 4 $$

Identify and sketch the quadric surface. $$ 9 x^{2}+4 y^{2}-36 z^{2}=0 $$

Identify and sketch the quadric surface. $$ 9 z^{2}-4 y^{2}-9 x^{2}=36 $$

Determine whether the statement is true or false. Explain your answer. Every plane has exactly two unit normal vectors.

Find \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\) $$ \mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}, \mathbf{v}=4 \mathbf{i}+\mathbf{j}-3 \mathbf{k}, \quad \mathbf{w}=\mathbf{j}+5 \mathbf{k} $$

Find unit vectors that satisfy the stated conditions. (a) Same direction as \(-\mathbf{i}+4 \mathbf{j}\). (b) Oppositely directed to \(6 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k} .\) (c) Same direction as the vector from the point \(A(-1,0,2)\) to the point \(B(3,1,1)\)

Find parametric equations of the line that satisfies the stated conditions. The line through \((-2,0,5)\) that is parallel to the line given by \(x=1+2 t, y=4-t, z=6+2 t\)

An equation is given in cylindrical coordinates. Express the equation in rectangular coordinates and sketch the graph. $$ z=r^{2} $$