Chapter 11

Find parametric equations for the lines. The line through the point \(P(3,-4,-1)\) parallel to the vector \(\mathbf{i}+\mathbf{j}+\mathbf{k}\)

Find a. \(\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proje \(\mathbf{u}\). $$\mathbf{v}=2 \mathbf{i}-4 \mathbf{j}+\sqrt{5} \mathbf{k}, \quad \mathbf{u}=-2 \mathbf{i}+4 \mathbf{j}-\sqrt{5} \mathbf{k}$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}-\mathbf{k}, \quad \mathbf{v}=\mathbf{i}-\mathbf{k}$$

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$3 \mathbf{u}$$

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x=2, y=3$$

Find parametric equations for the lines. The line through \(P(1,2,-1)\) and \(Q(-1,0,1)\)

Find a. \(\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proje \(\mathbf{u}\). $$\mathbf{v}=(3 / 5) \mathbf{i}+(4 / 5) \mathbf{k}, \quad \mathbf{u}=5 \mathbf{i}+12 \mathbf{j}$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=-\mathbf{i}+\mathbf{j}$$

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$-2 \mathbf{v}$$

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x=-1, \quad z=0$$

Find parametric equations for the lines. The line through \(P(-2,0,3)\) and \(Q(3,5,-2)\)

Find a. \(\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proje \(\mathbf{u}\). $$\mathbf{v}=10 \mathbf{i}+11 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{u}=3 \mathbf{j}+4 \mathbf{k}$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}, \quad \mathbf{v}=-\mathbf{i}+\mathbf{j}-2 \mathbf{k}$$

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$\mathbf{u}+\mathbf{v}$$

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

Find parametric equations for the lines. The line through \(P(1,2,0)\) and \(Q(1,1,-1)\)

Find a. \(\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proje \(\mathbf{u}\). $$\mathbf{v}=2 \mathbf{i}+10 \mathbf{j}-11 \mathbf{k}, \quad \mathbf{u}=2 \mathbf{i}+2 \mathbf{j}+\mathbf{k}$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=\mathbf{i}+\mathbf{j}-\mathbf{k}, \quad \mathbf{v}=\mathbf{0}$$

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$\mathbf{u}-\mathbf{v}$$

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x=1, \quad y=0$$

Find parametric equations for the lines. The line through the origin parallel to the vector \(2 \mathbf{j}+\mathbf{k}\)

Find a. \(\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proje \(\mathbf{u}\). $$\mathbf{v}=5 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=-3 \mathbf{j}$$

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$2\mathbf{u}-3\mathbf{v}$$

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

Find parametric equations for the lines. The line through the point (3,-2,1) parallel to the line \(x=1+2 t\) \(y=2-t, z=3 t\)

Find a. \(\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proje \(\mathbf{u}\). $$\mathbf{v}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{u}=\sqrt{2} \mathbf{i}+\sqrt{3} \mathbf{j}+2 \mathbf{k}$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=\mathbf{i} \times \mathbf{j}, \quad \mathbf{v}=\mathbf{j} \times \mathbf{k}$$

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$-2 \mathbf{u}+5 \mathbf{v}$$

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

Find parametric equations for the lines. The line through (1,1,1) parallel to the \(z\) -axis

Find a. \(\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proje \(\mathbf{u}\). $$\mathbf{v}=5 \mathbf{i}+\mathbf{j}, \quad \mathbf{u}=2 \mathbf{i}+\sqrt{17} \mathbf{j}$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=-8 \mathbf{i}-2 \mathbf{j}-4 \mathbf{k}, \quad \mathbf{v}=2 \mathbf{i}+2 \mathbf{j}+\mathbf{k}$$

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$\frac{3}{5}\mathbf{u}+\frac{4}{5} \mathbf{v}$$

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

Find parametric equations for the lines. The line through (2,4,5) perpendicular to the plane \(3 x+7 y\) \(-5 z=21\)

Find a. \(\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proje \(\mathbf{u}\). $$\mathbf{v}=\left\langle\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}\right\rangle, \quad \mathbf{u}=\left\langle\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{3}}\right\rangle$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=\frac{3}{2} \mathbf{i}-\frac{1}{2} \mathbf{j}+\mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}+2 \mathbf{k}$$

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$-\frac{5}{13}\mathbf{u}+\frac{12}{13}\mathbf{v}$$

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

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

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

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). 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. $$\mathbf{u}=\mathbf{i}, \quad \mathbf{v}=\mathbf{j}$$

Find the component form of the vector. The vector \(\overrightarrow{P Q}\), where \(P=(1,3)\) and \(Q=(2,-1)\).

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

Find parametric equations for the lines. The line through (2,3,0) perpendicular to the vectors \(\mathbf{u}=\mathbf{i}+\) \(2 \mathbf{j}+3 \mathbf{k}\) and \(\mathbf{v}=3 \mathbf{i}+4 \mathbf{j}+5 \mathbf{k}\)

Find the angles between the vectors in Exercises \(9-12\) to the nearest hundredth of a radian. $$\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}+\mathbf{k}, \quad \mathbf{v}=3 \mathbf{i}+4 \mathbf{k}$$

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. $$\mathbf{u}=\mathbf{i}-\mathbf{k}, \quad \mathbf{v}=\mathbf{j}$$

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. $$x^{2}+y^{2}+z^{2}=25, \quad y=-4$$