Chapter 11

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. $$(1,0,-1), \quad(0,3,0)$$

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

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. \(x^{2}+y^{2} \leq 1,\) no restriction on \(z\)

Sketch the surfaces PARABOLOIDS AND CONES $$z=x^{2}+4 y^{2}$$

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 \(\mathbf{w}\). $$\begin{array}{ccc} \mathbf{u} & \mathbf{v} & \mathbf{w} \\ \hline 2 \mathbf{i}+\mathbf{j} & 2 \mathbf{i}-\mathbf{j}+\mathbf{k} & \mathbf{i}+2 \mathbf{k} \end{array}$$

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

Show that the diagonals of a rhombus (parallelogram with sides of equal length) are perpendicular.

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(5 \mathbf{u}-\mathbf{v}\) if \(\mathbf{u}=\langle 1,1,-1\rangle\) and \(\mathbf{v}=\langle 2,0,3\rangle\)

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 PARABOLOIDS AND CONES $$z=8-x^{2}-y^{2}$$

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 \(\mathbf{w}\). $$\begin{array}{ccc} \mathbf{u} & \mathbf{v} & \mathbf{w} \\ \hline \mathbf{i}+\mathbf{j}-2 \mathbf{k} & -\mathbf{i}-\mathbf{k} & 2 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k} \end{array}$$

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

Show that squares are the only rectangles with perpendicular diagonals.

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(-2 \mathbf{u}+3 \mathbf{v}\) if \(\mathbf{u}=\langle-1,0,2\rangle\) and \(\mathbf{v}=\langle 1,1,1\rangle\)

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. a. \(x=y, z=0\) b. \(x=y, \quad\) no restriction on \(z\)

Sketch the surfaces PARABOLOIDS AND CONES $$x=4-4 y^{2}-z^{2}$$

Let \(\mathbf{u}=5 \mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{v}=\) \(\mathbf{j}-5 \mathbf{k}, \mathbf{w}=-15 \mathbf{i}+3 \mathbf{j}-3 \mathbf{k} .\) Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

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 \((1,1,-1),(2,0,2),\) and (0,-2,1)

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

Sketch the surfaces PARABOLOIDS AND CONES $$y=1-x^{2}-z^{2}$$

Let \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}\) \(\mathbf{v}=-\mathbf{i}+\mathbf{j}+\mathbf{k}, \quad \mathbf{w}=\mathbf{i}+\mathbf{k}, \quad \mathbf{r}=-(\pi / 2) \mathbf{i}-\pi \mathbf{j}+(\pi / 2) \mathbf{k}\) Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

Find equations for the planes. The plane through \((2,4,5),(1,5,7),\) and (-1,6,8)

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

Sketch the surfaces PARABOLOIDS AND CONES $$x^{2}+y^{2}=z^{2}$$

A gun with muzzle velocity of \(1200 \mathrm{ft} / \mathrm{sec}\) is fired at an angle of \(8^{\circ}\) above the horizontal. Find the horizontal and vertical components of the velocity.

Find equations for the planes. The plane through \(P_{0}(2,4,5)\) perpendicular to the line $$x=5+t, \quad y=1+3 t, \quad z=4 t$$

Express each vector as a product of its length and direction. $$2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}$$

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(1,1,1), \quad P_{2}(3,3,0)$$

Sketch the surfaces PARABOLOIDS AND CONES $$4 x^{2}+9 z^{2}=9 y^{2}$$

Find equations for the planes. The plane through \(A(1,-2,1)\) perpendicular to the vector from the origin to \(A\)

Express each vector as a product of its length and direction. $$9 \mathbf{i}-2 \mathbf{j}+6 \mathbf{k}$$

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(-1,1,5), \quad P_{2}(2,5,0)$$

Which of the following are always true, and which are not always true? Give reasons for your answers. a. \(|\mathbf{u}|=\sqrt{\mathbf{u} \cdot \mathbf{u}}\) b. \(\mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|\) c. \(\mathbf{u} \times \mathbf{0}=\mathbf{0} \times \mathbf{u}=\mathbf{0}\) d. \(\mathbf{u} \times(-\mathbf{u})=\mathbf{0}\) e. \(\mathbf{u} \times \mathbf{v}=\mathbf{v} \times \mathbf{u}\) f. \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=\mathbf{u} \times \mathbf{v}+\mathbf{u} \times \mathbf{w}\) g. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{v}=0\) h. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\)

Sketch the surfaces HYPERBOLOIDS $$x^{2}+y^{2}-z^{2}=1$$

a. Cauchy-Schwarz inequality since \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) show that the inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) holds for any vectors \(\mathbf{u}\) and \(\mathbf{v}\). b. Under what circumstances, if any, does \(|\mathbf{u} \cdot \mathbf{v}|\) equal \(|\mathbf{u}||\mathbf{v}| ?\) Give reasons for your answer.

Find the point of intersection of the lines \(x=2 t+1\) \(y=3 t+2, \quad z=4 t+3,\) and \(x=s+2, y=2 s+4, z=\) \(-4 s-1,\) and then find the plane determined by these lines.

Express each vector as a product of its length and direction. $$5 \mathbf{k}$$

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(1,4,5), \quad P_{2}(4,-2,7)$$

Which of the following are always true, and which are not always true? Give reasons for your answers. a. \(\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}\) b. \(\mathbf{u} \times \mathbf{v}=-(\mathbf{v} \times \mathbf{u})\) c. \((-\mathbf{u}) \times \mathbf{v}=-(\mathbf{u} \times \mathbf{v})\) d. \((c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})=c(\mathbf{u} \cdot \mathbf{v}) \quad \text { (any number } c)\) e. \(c(\mathbf{u} \times \mathbf{v})=(c \mathbf{u}) \times \mathbf{v}=\mathbf{u} \times(c \mathbf{v}) \quad \text { (any number } c)\) f. \(\mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|^{2}\) g. \((\mathbf{u} \times \mathbf{u}) \cdot \mathbf{u}=0\) h. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u}=\mathbf{v} \cdot(\mathbf{u} \times \mathbf{v})\)

Sketch the surfaces HYPERBOLOIDS $$y^{2}+z^{2}-x^{2}=1$$

Dot multiplication is positive definite Show that dot multiplication of vectors is positive definite; that is, show that \(\mathbf{u} \cdot \mathbf{u} \geq \mathbf{0}\) for every vector \(\mathbf{u}\) and that \(\mathbf{u} \cdot \mathbf{u}=0\) if and only if \(\mathbf{u}=\mathbf{0}\)

Find the point of intersection of the lines \(x=t, y=-t+2\) \(z=t+1,\) and \(x=2 s+2, y=s+3, z=5 s+6,\) and then find the plane determined by these lines.

Express each vector as a product of its length and direction. $$\frac{3}{5} \mathbf{i}+\frac{4}{5} \mathbf{k}$$

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(3,4,5), \quad P_{2}(2,3,4)$$

Given nonzero vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w},\) use dot product and cross product notation, as appropriate, to describe the following. a. The vector projection of \(\mathbf{u}\) onto \(\mathbf{v}\) b. A vector orthogonal to \(\mathbf{u}\) and \(\mathbf{v}\) c. A vector orthogonal to \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{w}\) d. The volume of the parallelepiped determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) e. A vector orthogonal to \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{u} \times \mathbf{w}\) f. A vector of length \(|\mathbf{u}|\) in the direction of \(\mathbf{v}\)

Sketch the surfaces HYPERBOLOIDS $$z^{2}-x^{2}-y^{2}=1$$

Orthogonal unit vectors If \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are orthogonal unit vectors and \(\mathbf{v}=a \mathbf{u}_{1}+b \mathbf{u}_{2},\) find \(\mathbf{v} \cdot \mathbf{u}_{1}\)

Find the plane containing the intersecting lines. \(\begin{array}{ll}L 1: x=-1+t, & y=2+t, z=1-t ; \quad-\infty

Express each vector as a product of its length and direction. $$\frac{1}{\sqrt{6}} \mathbf{i}-\frac{1}{\sqrt{6}} \mathbf{j}-\frac{1}{\sqrt{6}} \mathbf{k}$$