91Ó°ÊÓ

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. a. \(x^{2}+y^{2} \leq 1, \quad z=0\) b. \(x^{2}+y^{2} \leq 1, \quad z=3\) c. \(x^{2}+y^{2} \leq 1, \quad\) no restriction on \(z\)

Show that squares are the only rectangles with perpendicular diagonals.

Sketch the surfaces. $$9 x^{2}+4 y^{2}+36 z^{2}=36$$

Express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k}\). \(\overrightarrow{A B}\) if \(A\) is the point (1,0,3) and \(B\) is the point (-1,4,5)

Verify that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}=\) \((\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}\) and find the volume of the parallelepiped (box) determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}.\)

Find equations for the planes The plane through \(P_{0}(0,2,-1)\) normal to \(\mathbf{n}=3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}\)

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. a. \(1 \leq x^{2}+y^{2}+z^{2} \leq 4\) b. \(x^{2}+y^{2}+z^{2} \leq 1, \quad z \geq 0\)

Sketch the surfaces. $$z=x^{2}+4 y^{2}$$

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

Verify that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}=\) \((\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}\) and find the volume of the parallelepiped (box) determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}.\)