91Ó°ÊÓ

Use traces to sketch and identify the surface. $$ z^{2}-4 x^{2}-y^{2}=4 $$

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. $$ A(0,6,-1), \quad B(3,4,4) $$

Is the line through \((-2,4,0)\) and \((1,1,1)\) perpendicular to the line through \((2,3,4)\) and \((3,-1,-8) ?\)

A street vendor sells \(a\) hamburgers, \(b\) hot dogs, and \(c\) soft drinks on a given day. He charges 4 dollar for a hamburger, 2.50 dollar for a hot dog, and 1 dollar for a soft drink. If \(A=langle a, b, c\rangle\) and \(P=langle 4,2.5,1rangle,\) what is the meaning of the dot product \(A cdot P ?\)

Use traces to sketch and identify the surface. $$ 9 y^{2}+4 z^{2}=x^{2}+36 $$

Find the sum of the given vectors and illustrate geometrically. $$ \langle- 1,4\rangle, \quad\langle 6,-2\rangle $$

Find an equation of the sphere that passes through the point \((4,3,-1)\) and has center \((3,8,1) .\)

(a) Find symmetric equations for the line that passes through the point \((1,-5,6)\) and is parallel to the vector \(\langle- 1,2,-3\rangle .\) (b) Find the points in which the required line in part (a) intersects the coordinate planes.

\(15-20\) Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $$ \mathbf{a}=\langle 4,3\rangle, \quad \mathbf{b}=\langle 2,-1\rangle $$

Find an equation of the sphere that passes through the origin and whose center is \((1,2,3) .\)