91Ó°ÊÓ

If \(\mathbf{a}=\mathbf{i}-2 \mathbf{k}\) and \(\mathbf{b}=\mathbf{j}+\mathbf{k},\) find \(\mathbf{a} \times \mathbf{b} .\) Sketch \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{a} \times \mathbf{b}\) as vectors starting at the origin.

(a) Find and identify the traces of the quadric surface \(x^{2}+y^{2}-z^{2}=1\) and explain why the graph looks like the graph of the hyperboloid of one sheet in Table \(1 .\) (b) If we change the equation in part (a) to \(x^{2}-y^{2}+z^{2}=1\) how is the graph affected? (c) What if we change the equation in part (a) to \(x^{2}+y^{2}+2 y-z^{2}=0 ?\)

Find parametric equations and symmetric equations for the line. The line through the points \((-8,1,4)\) and \((3,-2,4)\)

Find the lengths of the sides of the triangle \(P Q R .\) Is it a right triangle? Is it an isosceles triangle? $$ P(3,-2,-3), \quad Q(7,0,1), \quad R(1,2,1) $$

\(9-12\) Find the vector, not with determinants, but by using properties of cross products. $$ (\mathbf{i} \times \mathbf{j}) \times \mathbf{k} $$

Find \(\mathbf{a} \cdot \mathbf{b}\) $$ |\mathbf{a}|=7, \quad|\mathbf{b}|=4, \quad \text { the angle between a and } \mathbf{b} \text { is } 30^{\circ} $$

Find a vector a with representation given by the directed line segment \(\overrightarrow{A B}\). Draw \(\overrightarrow{A B}\) and the equivalent representation starting at the origin. $$ A(-2,1), \quad B(1,2) $$

Find a vector a with representation given by the directed line segment \(\overrightarrow{A B}\). Draw \(\overrightarrow{A B}\) and the equivalent representation starting at the origin. $$ A(-5,-1), \quad B(-3,3) $$

Find the lengths of the sides of the triangle \(P Q R .\) Is it a right triangle? Is it an isosceles triangle? $$ P(2,-1,0), \quad Q(4,1,1), \quad R(4,-5,4) $$

Find \(\mathbf{a} \cdot \mathbf{b}\) $$ |\mathbf{a}|=80, \quad|\mathbf{b}|=50, \quad \text { the angle between a and } \mathbf{b} \text { is } 3 \pi / 4 $$