Chapter 11

Find parametric equations for the lines. The \(x\) -axis

Sketch the coordinate axes and then include the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u} \times \mathbf{v}\) as vectors starting at the origin. $$\mathbf{u}=\mathbf{i}-\mathbf{k}, \quad \mathbf{v}=\mathbf{j}+\mathbf{k}$$

Find the component form of the vector. The vector from the point \(A=(2,3)\) to the origin.

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}+(z+3)^{2}=25, \quad z=0$$

Find parametric equations for the lines. The z-axis

Find the angles between the vectors in Exercises \(9-12\) to the nearest hundredth of a radian. $$\mathbf{u}=\mathbf{i}+\sqrt{2} \mathbf{j}-\sqrt{2} \mathbf{k}, \quad \mathbf{v}=-\mathbf{i}+\mathbf{j}+\mathbf{k}$$

Sketch the coordinate axes and then include the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u} \times \mathbf{v}\) as vectors starting at the origin. $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=\mathbf{i}+2 \mathbf{j}$$

Find the component form of the vector. The sum of \(\overrightarrow{A B}\) and \(\overrightarrow{C D},\) where \(A=(1,-1), B=(2,0)\) \(C=(-1,3),\) and \(D=(-2,2)\).

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+(y-1)^{2}+z^{2}=4, \quad y=0$$

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

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. $$(0,0,0), \quad(1,1,3 / 2)$$

Triangle Find the measures of the angles of the triangle whose vertices are \(A=(-1,0), B=(2,1),\) and \(C=(1,-2)\)

Sketch the coordinate axes and then include the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u} \times \mathbf{v}\) as vectors starting at the origin. $$\mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j}$$

Find the component form of the vector. The unit vector that makes an angle \(\theta=2 \pi / 3\) with the positive \(x\) -axis.

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}=4, \quad z=y$$

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

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

Rectangle Find the measures of the angles between the diagonals of the rectangle whose vertices are \(A=(1,0), B=(0,3)\) \(C=(3,4),\) and \(D=(4,1)\)

Sketch the coordinate axes and then include the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u} \times \mathbf{v}\) as vectors starting at the origin. $$\mathbf{u}=\mathbf{j}+2 \mathbf{k}, \quad \mathbf{v}=\mathbf{i}$$

Find the component form of the vector. The unit vector that makes an angle \(\theta=-3 \pi / 4\) with the positive \(x\) -axis.

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}+z^{2}=4, \quad y=x$$

Sketch the surfaces CYLINDERS $$x^{2}+4 z^{2}=16$$

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

Direction angles and direction cosines The direction angles \(\alpha, \beta,\) and \(\gamma\) of a vector \(\mathbf{v}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) are defined as follows: \(\alpha\) is the angle between \(\mathbf{v}\) and the positive \(x\) -axis \((0 \leq \alpha \leq \pi)\) \(\beta\) is the angle between \(v\) and the positive \(y\) -axis \((0 \leq \beta \leq \pi)\) \(\gamma\) is the angle between \(\mathbf{v}\) and the positive \(z\) -axis \((0 \leq \gamma \leq \pi)\) (graph cannot copy) a. Show that $$ \cos \alpha=\frac{a}{|\mathbf{v}|}, \quad \cos \beta=\frac{b}{|\mathbf{v}|}, \quad \cos \gamma=\frac{c}{|\mathbf{v}|} $$ and \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 .\) These cosines are called the direction cosines of \(\mathbf{v}\). b. Unit vectors are built from direction cosines Show that if \(\mathbf{v}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) is a unit vector, then \(a, b,\) and \(c\) are the direction cosines of \(\mathbf{v}\)

a. Find the area of the triangle determined by the points \(P, Q\) and \(R\). b. Find a unit vector perpendicular to plane \(P Q R\). $$P(1,-1,2), \quad Q(2,0,-1), \quad R(0,2,1)$$

Find the component form of the vector. The unit vector obtained by rotating the vector \langle 0,1\rangle \(120^{\circ}\) counterclockwise about the origin.

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

Sketch the surfaces CYLINDERS $$4 x^{2}+y^{2}=36$$

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

a. Find the area of the triangle determined by the points \(P, Q\) and \(R\). b. Find a unit vector perpendicular to plane \(P Q R\). $$P(1,1,1), \quad Q(2,1,3), \quad R(3,-1,1)$$

Find the component form of the vector. The unit vector obtained by rotating the vector \langle 1,0\rangle \(135^{\circ}\) counterclockwise about the origin.

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$z=y^{2}, \quad x=1$$

Sketch the surfaces ELLIPSOIDS $$9 x^{2}+y^{2}+z^{2}=9$$

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

Find the acute angle between the given lines by using vectors parallel to the lines. $$y=x, \quad y=2 x+3$$

a. Find the area of the triangle determined by the points \(P, Q\) and \(R\). b. Find a unit vector perpendicular to plane \(P Q R\). $$P(2,-2,1), \quad Q(3,-1,2), \quad R(3,-1,1)$$

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(\overrightarrow{P_{1} \vec{P}_{2}}\) if \(P_{1}\) is the point (5,7,-1) and \(P_{2}\) is the point (2,9,-2)

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

Sketch the surfaces ELLIPSOIDS $$4 x^{2}+4 y^{2}+z^{2}=16$$

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. $$(0,2,0), \quad(3,0,0)$$

Find the acute angle between the given lines by using vectors parallel to the lines. $$2-x+2 y=0, \quad 3 x-4 y=-12$$

a. Find the area of the triangle determined by the points \(P, Q\) and \(R\). b. Find a unit vector perpendicular to plane \(P Q R\). $$P(-2,2,0), \quad Q(0,1,-1), \quad R(-1,2,-2)$$

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

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. a. \(0 \leq x \leq 1\) b. \(0 \leq x \leq 1, \quad 0 \leq y \leq 1\) c. \(0 \leq x \leq 1, \quad 0 \leq y \leq 1, \quad 0 \leq z \leq 1\)

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

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. $$(2,0,2), \quad(0,2,0)$$

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 (-7,-8,1) and \(B\) is the point (-10,8,1)

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

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

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