91Ó°ÊÓ

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\) $$\mathbf{u}=\mathbf{i}+\mathbf{j}-\mathbf{k}, \quad \mathbf{v}=\mathbf{0}$$

Find parametric equations for the lines. The line through (0,-7,0) perpendicular to the plane \(x+2 y+2 z=13\)

Find the component form of the vector. The vector \(\overrightarrow{O P}\) where \(O\) is the origin and \(P\) is the midpoint of segment \(R S\), where \(R=(2,-1)\) and \(S=(-4,3)\)

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$y=x^{2}, \quad z=0$$

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. $$(1,0,0), \quad(1,1,0)$$

Show that squares are the only rectangles with perpendicular diagonals.

Prove that a parallelogram is a rectangle if and only if its diagonals are equal in length. (This fact is often exploited by carpenters.)

Find equations for the planes. The plane through (1,-1,3) parallel to the plane $$3 x+y+z=7$$

Verify that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}=\) \((w \times u) \cdot v\) and find the volume of the parallelepiped (box) determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) (IMAGE CANNOT COPY) $$\mathbf{i}+\mathbf{j}-\mathbf{2 k} \quad-\mathbf{i}-\mathbf{k} \quad 2 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k}$$

The plane perpendicular to the a. \(x\) -axis at (3,0,0) b. \(y\) -axis at (0,-1,0) c. \(z\) -axis at (0,0,-2)