91Ó°ÊÓ

Identify and sketch the quadric surface. \(v^{2} \quad q^{2}\) $$ \frac{x^{2}}{4}+\frac{y^{2}}{9}-\frac{z^{2}}{16}=1 $$

Determine whether the statement is true or false. Explain your answer. If two distinct vectors \(\mathbf{v}\) and \(\mathbf{w}\) are drawn with the same initial point, then a vector drawn between the terminal points of \(\mathbf{v}\) and \(\mathbf{w}\) will be either \(\mathbf{v}-\mathbf{w}\) or \(\mathbf{w}-\mathbf{v}\).

Find parametric equations of the line that satisfies the stated conditions. The line that is tangent to the parabola \(y=x^{2}\) at the point \((-2,4)\)

Find the area of the parallelogram that has \(\mathbf{u}\) and \(\mathbf{v}\) as adjacent sides. $$ \mathbf{u}=2 \mathbf{i}+3 \mathbf{j}, \mathbf{v}=-\mathbf{i}+2 \mathbf{j}-2 \mathbf{k} $$

Identify and sketch the quadric surface. \(v^{2} \quad q^{2}\) $$ x^{2}+y^{2}-z^{2}=9 $$

Determine whether the line and plane intersect; if so, find the coordinates of the intersection. $$ \begin{aligned} &\text { (a) } x=3 t, y=5 t, z=-t \\ &\quad 2 x-y+z+1=0 \\ &\text { (b) } x=1+t, y=-1+3 t, z=2+4 t \\ &\quad x-y+4 z=7 \end{aligned} $$

A sphere has center in the first octant and is tangent to each of the three coordinate planes. The distance from the origin to the sphere is \(3-\sqrt{3}\) units. Find an equation for the sphere.

Determine whether the statement is true or false. Explain your answer. There are exactly two unit vectors that are parallel to a given nonzero vector.

An equation is given in cylindrical coordinates. Express the equation in rectangular coordinates and sketch the graph. $$ r=3 $$

Identify and sketch the quadric surface. \(v^{2} \quad q^{2}\) $$ 4 z^{2}=x^{2}+4 y^{2} $$