91Ó°ÊÓ

Describe and sketch the surface. $$ 4 x^{2}+y^{2}=4 $$

Find a vector equation and parametric equations for the line. The line through the point \((0,14,-10)\) and parallel to the line \(x=-1+2 t, y=6-3 t, z=3+9 t\)

Describe and sketch the surface. $$ z=1-y^{2} $$

Find a vector equation and parametric equations for the line. The line through the point \((1,0,6)\) and perpendicular to the plane \(x+3 y+z=5\)

What does the equation \(x=4\) represent in \(\mathbb{R}^{2} ?\) What does it represent in \(\mathbb{R}^{3}\) ? Illustrate with sketches.

Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both a and \(\mathbf{b} .\) $$ \mathbf{a}=\frac{1}{2} \mathbf{i}+\frac{1}{3} \mathbf{j}+\frac{1}{4} \mathbf{k}, \quad \mathbf{b}=\mathbf{i}+2 \mathbf{j}-3 \mathbf{k} $$

Find \(\mathbf{a} \cdot \mathbf{b}\) $$ \mathbf{a}=\left\langle 4,1, \frac{1}{4}\right\rangle, \quad \mathbf{b}=\langle 6,-3,-8\rangle $$

What does the equation \(y=3\) represent in \(\mathbb{R}^{3} ?\) What does \(z=5\) represent? What does the pair of equations \(y=3, z=5\) represent? In other words, describe the set of points \((x, y, z)\) such that \(y=3\) and \(z=5 .\) Illustrate with a sketch.

\(6-12\) Find parametric equations and symmetric equations for the line. The line through the origin and the point \((4,3,-1)\)

Find \(\mathbf{a} \cdot \mathbf{b}\) $$ \mathbf{a}=\langle p,-p, 2 p\rangle, \quad \mathbf{b}=\langle 2 q, q,-q\rangle $$