Chapter 12

Find parametric equations for the lines in Exercises 1–12. The line through the point \(P(3,-4,-1)\) parallel to the vector \(\mathbf{i}+\mathbf{j}+\mathbf{k}\)

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u} .\) $$ \mathbf{u}=2 \mathbf{i}-2 \mathbf{j}-\mathbf{k}, \quad \mathbf{v}=\mathbf{i}-\mathbf{k} $$

In Exercises \(1-8,\) find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) 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 projv \(\mathbf{u}\) . $$ \mathbf{v}=2 \mathbf{i}-4 \mathbf{j}+\sqrt{5} \mathbf{k}, \quad \mathbf{u}=-2 \mathbf{i}+4 \mathbf{j}-\sqrt{5} \mathbf{k} $$

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ 3 \mathbf{u} $$

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ x=2, \quad y=3 $$

Find parametric equations for the lines in Exercises 1–12. The line through \(P(1,2,-1)\) and \(Q(-1,0,1)\)

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u} .\) $$ \mathbf{u}=2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=-\mathbf{i}+\mathbf{j} $$

In Exercises \(1-8,\) find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) 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 projv \(\mathbf{u}\) . $$ \mathbf{v}=(3 / 5) \mathbf{i}+(4 / 5) \mathbf{k}, \quad \mathbf{u}=5 \mathbf{i}+12 \mathbf{j} $$

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ -2 \mathbf{v} $$

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ x=-1, \quad z=0 $$

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a ) \(-(1)\) . $$ z^{2}+4 y^{2}-4 x^{2}=4 $$

Find parametric equations for the lines in Exercises 1–12. The line through \(P(-2,0,3)\) and \(Q(3,5,-2)\)

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u} .\) $$ \mathbf{u}=2 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}, \quad \mathbf{v}=-\mathbf{i}+\mathbf{j}-2 \mathbf{k} $$

In Exercises \(1-8,\) find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) 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 projv \(\mathbf{u}\) . $$ \mathbf{v}=10 \mathbf{i}+11 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{u}=3 \mathbf{j}+4 \mathbf{k} $$

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ \mathbf{u}+\mathbf{v} $$

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ y=0, \quad z=0 $$

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a ) \(-(1)\) . $$ y^{2}+z^{2}=x^{2} $$

Find parametric equations for the lines in Exercises 1–12. The line through \(P(1,2,0)\) and \(Q(1,1,-1)\)

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

In Exercises \(1-8,\) find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) 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 projv \(\mathbf{u}\) . $$ \mathbf{v}=2 \mathbf{i}+10 \mathbf{j}-11 \mathbf{k}, \quad \mathbf{u}=2 \mathbf{i}+2 \mathbf{j}+\mathbf{k} $$

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ \mathbf{u}-\mathbf{v} $$

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ x=1, \quad y=0 $$

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a ) \(-(1)\) . $$ x=y^{2}-z^{2} $$

Find parametric equations for the lines in Exercises 1–12. The line through the origin parallel to the vector \(2 \mathbf{j}+\mathbf{k}\)

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u} .\) $$ \mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=-3 \mathbf{j} $$

In Exercises \(1-8,\) find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) 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 projv \(\mathbf{u}\) . $$ \mathbf{v}=5 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k} $$

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ x^{2}+y^{2}=4, \quad z=0 $$

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ 2 \mathbf{u}-3 \mathbf{v} $$

Find parametric equations for the lines in Exercises 1–12. The line through the point \((3,-2,1)\) parallel to the line \(x=1+2 t, y=2-t, z=3 t\)

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

In Exercises \(1-8,\) find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) 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 projv \(\mathbf{u}\) . $$ \mathbf{v}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{u}=\sqrt{2} \mathbf{i}+\sqrt{3} \mathbf{j}+2 \mathbf{k} $$

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ -2 \mathbf{u}+5 \mathbf{v} $$

In Exercises \(1-12,\) 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 in Exercises 1–12. The line through \((1,1,1)\) parallel to the \(z\) -axis

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u} .\) $$ \mathbf{u}=-8 \mathbf{i}-2 \mathbf{j}-4 \mathbf{k}, \quad \mathbf{v}=2 \mathbf{i}+2 \mathbf{j}+\mathbf{k} $$

In Exercises \(1-8,\) find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) 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 projv \(\mathbf{u}\) . $$ \mathbf{v}=5 \mathbf{i}+\mathbf{j}, \quad \mathbf{u}=2 \mathbf{i}+\sqrt{17} \mathbf{j} $$

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ \frac{3}{5} \mathbf{u}+\frac{4}{5} \mathbf{v} $$

In Exercises \(1-12,\) 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 in Exercises 1–12. The line through \((2,4,5)\) perpendicular to the plane \(3 x+7 y-5 z=21\)

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u} .\) $$ \mathbf{u}=\frac{3}{2} \mathbf{i}-\frac{1}{2} \mathbf{j}+\mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}+2 \mathbf{k} $$

In Exercises \(1-8,\) find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) 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 projv \(\mathbf{u}\) . $$ \mathbf{v}=\left\langle\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}\right\rangle, \quad \mathbf{u}=\left\langle\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{3}}\right\rangle $$

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ y^{2}+z^{2}=1, \quad x=0 $$

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ -\frac{5}{13} \mathbf{u}+\frac{12}{13} \mathbf{v} $$

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a ) \(-(1)\) . $$ z^{2}+x^{2}-y^{2}=1 $$

Find parametric equations for the lines in Exercises 1–12. The line through \((0,-7,0)\) perpendicular to the plane \(x+2 y+2 z=13\)

In Exercises \(9-14,\) 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 angles between the vectors in Exercises \(9-12\) to the nearest hundredth of a radian. $$ \mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}+2 \mathbf{j}-\mathbf{k} $$

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ x^{2}+y^{2}+z^{2}=1, \quad x=0 $$

In Exercises \(9-16,\) find the component form of the vector. The vector \(\overrightarrow{P Q},\) where \(P=(1,3)\) and \(Q=(2,-1)\)

Find parametric equations for the lines in Exercises 1–12. The line through \((2,3,0)\) perpendicular to the vectors \(\mathbf{u}=\mathbf{i}+ 2 \mathbf{j}+3 \mathbf{k}\) and \(\mathbf{v}=3 \mathbf{i}+4 \mathbf{j}+5 \mathbf{k}\)