91Ó°ÊÓ

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (2,-7,3),(-1,5,8),(4,6,-1) $$

Determine whether the lines intersect, and if so, find the point of intersection and the cosine of the angle of intersection.\(x=2 t-1, y=-4 t+10, z=t\) \(x=-5 s-12, y=3 s+11, z=-2 s-4\)

find the area of the triangle with the given vertices. (Hint: \(\frac{1}{2}\|\mathbf{u} \times \mathbf{v}\|\) is the area of the triangle having \(\mathbf{u}\) and \(\mathbf{v}\) as adjacent sides. $$ (0,0,0),(1,2,3),(-3,0,0) $$

Convert the point from spherical coordinates to rectangular coordinates. \((12,3 \pi / 4, \pi / 9)\)

Complete the square to write the equation of the sphere in standard form. Find the center and radius. \(9 x^{2}+9 y^{2}+9 z^{2}-6 x+18 y+1=0\)

Find an equation for the surface of revolution generated by revolving the curve in the indicated coordinate plane about the given axis. Equation of Curve \(\quad\) Coordinate Plane \(\quad\) Axis of Revolution \(z=3 y \quad y z\) -plane \(\quad y\) -axis

State the geometric properties of the cross product.

If the magnitudes of two vectors are doubled, how will the magnitude of the cross product of the vectors change? Explain.

Determine which of the following are defined for nonzero vectors \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w} .\) Explain your reasoning. (a) \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})\) (b) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\) (c) \(\mathbf{u} \cdot \mathbf{v}+\mathbf{w}\) (d) \(\|\mathbf{u}\| \cdot(\mathbf{v}+\mathbf{w})\)

What is meant by the trace of a surface? How do you find a trace?