Chapter 10

A description of a line is given. Find parametric equations for the line. The line parallel to the \(y\) -axis that crosses the \(x z\) -plane where \(x=-3\) and \(z=2\)

(a) Calculate proj, \(\mathbf{u}\) . (b) Resolve \(\mathbf{u}\) into \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v} .\) $$ \mathbf{u}=\langle 7,-4\rangle, \quad \mathbf{v}=\langle 2,1\rangle $$

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=\langle 3,0,-4\rangle, \quad \mathbf{b}=\langle 1,1,1\rangle, \quad \mathbf{c}=\langle 7,4,0\rangle $$

\(27-30\) Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\) . $$ \mathbf{u}=\langle 0,-5\rangle $$

Determine whether or not the given vectors are perpendicular. $$ \langle 0.3,1.2,-0.9\rangle,\langle 10,-5,10\rangle $$

A description of a plane is given. Find an equation for the plane. The plane that crosses the \(x\) -axis where \(x=1,\) the \(y\) -axis where \(y=3,\) and the \(z\) -axis where \(z=4\)

(a) Calculate proj, \(\mathbf{u}\) . (b) Resolve \(\mathbf{u}\) into \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v} .\) $$ \mathbf{u}=\langle 1,2\rangle, \quad \mathbf{v}=\langle 1,-3\rangle $$

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=\langle 2,3,-2\rangle, \quad \mathbf{b}=\langle- 1,4,0\rangle, \quad \mathbf{c}=\langle 3,-1,3\rangle $$

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\langle 2,7\rangle, \quad \mathbf{v}=\langle 3,1\rangle $$

Determine whether or not the given vectors are perpendicular. $$ \langle x,-2 x, 3 x\rangle,\langle 5,7,3\rangle $$

A description of a plane is given. Find an equation for the plane. The plane that crosses the \(x\) -axis where \(x=-2,\) the \(y\) -axis where \(y=-1,\) and the \(z\) -axis where \(z=3\)

(a) Calculate proj, \(\mathbf{u}\) . (b) Resolve \(\mathbf{u}\) into \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v} .\) $$ \mathbf{u}=\langle 11,3\rangle, \quad \mathbf{v}=\langle- 3,-2\rangle $$

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=\langle 1,-1,0\rangle, \quad \mathbf{b}=\langle- 1,0,1\rangle, \quad \mathbf{c}=\langle 0,-1,1\rangle $$

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\langle- 2,5\rangle, \quad \mathbf{v}=\langle 2,-8\rangle $$

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $$ \mathbf{u}=\langle 2,-2,-1\rangle, \quad \mathbf{v}=\langle 1,2,2\rangle $$

A description of a plane is given. Find an equation for the plane. The plane that is parallel to the plane \(x-2 y+4 z=6\) and contains the origin.

(a) Calculate proj, \(\mathbf{u}\) . (b) Resolve \(\mathbf{u}\) into \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v} .\) $$ \mathbf{u}=\langle 2,9\rangle, \quad \mathbf{v}=\langle- 3,4\rangle $$

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=\mathbf{i}-\mathbf{j}+\mathbf{k}, \quad \mathbf{b}=-\mathbf{j}+\mathbf{k}, \quad \mathbf{c}=\mathbf{i}+\mathbf{j}+\mathbf{k} $$

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\langle 0,-1\rangle, \quad \mathbf{v}=\langle- 2,0\rangle $$

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $$ \mathbf{u}=\langle 4,0,2\rangle, \quad \mathbf{v}=\langle 2,-1,0\rangle $$

A description of a plane is given. Find an equation for the plane. The plane that contains the line \(x=1-t, y=2+t\) \(z=-3 t\) and the point \(P(2,0,-6) .\) [Hint: A vector from any point on the line to \(P\) will lie in the plane. \(]\)

(a) Calculate proj, \(\mathbf{u}\) . (b) Resolve \(\mathbf{u}\) into \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v} .\) $$ \mathbf{u}=\langle 1,1\rangle, \quad \mathbf{v}=\langle 2,-1\rangle $$

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{c}=6 \mathbf{i} $$

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\mathbf{i}, \quad \mathbf{v}=-2 \mathbf{j} $$

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $$ \mathbf{u}=\mathbf{j}+\mathbf{k}, \quad \mathbf{v}=\mathbf{i}+2 \mathbf{j}-3 \mathbf{k} $$

Intersection of a Line and a Plane A line has parametric equations $$ x=2+t, \quad y=3 t, \quad z=5-t $$ and a plane has equation \(5 x-2 y-2 z=1\) (a) For what value of \(t\) does the corresponding point on the line intersect the plane? (b) At what point do the line and the plane intersect?

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q .\) $$ \mathbf{F}=4 \mathbf{i}-5 \mathbf{j} ; \quad P(0,0), Q(3,8) $$

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j} $$

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $$ \mathbf{u}=\mathbf{i}+2 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{v}=4 \mathbf{i}-3 \mathbf{k} $$

Lines and Planes A line is parallel to the vector \(\mathbf{v},\) and a plane has normal vector \(\mathbf{n}\) . (a) If the line is perpendicular to the plane, what is the relationship between \(\mathbf{v}\) and \(\mathbf{n}\) (parallel or perpendicular)? (b) If the line is parallel to the plane (that is, the line and the plane do not intersect), what is the relationship between \(\mathbf{v}\) and \(\mathbf{n}(\text { parallel or perpendicular) } ?\) (c) Parametric equations for two lines are given. Which line is parallel to the plane \(x-y+4 z=6 ?\) Which line is perpendicular to this plane? Line \(1 : \quad x=2 t, \quad y=3-2 t, \quad z=4+8 t\) Line \(2 : \quad x=-2 t, \quad y=5+2 t, \quad z=3+t\)

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q .\) $$ \mathbf{F}=400 \mathbf{i}+50 \mathbf{j} ; \quad P(-1,1), Q(200,1) $$

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j} $$

Find the direction angles of the given vector, rounded to the nearest degree. $$ 3 \mathbf{i}+4 \mathbf{j}+5 \mathbf{k} $$

Same Line: Different Parametric Equations Every line can be described by infinitely many different sets of parametric equations, since any point on the line and any vector parallel to the line can be used to construct the equations. But how can we tell whether two sets of parametric equations rep- resent the same line? Consider the following two sets of para- metric equations: Line \(1 : \quad x=1-t, \quad y=3 t, \quad z=-6+5 t\) Line \(2 : \quad x=-1+2 t, \quad y=6-6 t, \quad z=4-10 t\) (a) Find two points that lie on Line 1 by setting \(t=0\) and \(t=1\) in its parametric equations. Then show that these points also lie on Line 2 by finding two values of the parameter that give these points when substituted into the parametric equations for Line 2 . (b) Show that the following two lines are not the same by finding a point on Line 3 and then showing that it does not lie on Line \(4 .\) Line \(3 : \quad x=4 t, \quad y=3-6 t, \quad z=-5+2 t\) Line \(4 : \quad x=8-2 t, \quad y=-9+3 t, \quad z=6-t\)

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q .\) $$ \mathbf{F}=10 \mathbf{i}+3 \mathbf{j} ; \quad P(2,3), Q(6,-2) $$

Order of Operations in the Triple Product Given three vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w},\) their scalar triple product can be performed in six different orders: $$ \begin{array}{ll}{\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}),} & {\mathbf{u} \cdot(\mathbf{w} \times \mathbf{v}), \quad \mathbf{v} \cdot(\mathbf{u} \times \mathbf{w})} \\ {\mathbf{v} \cdot(\mathbf{w} \times \mathbf{u}),} & {\mathbf{w} \cdot(\mathbf{u} \times \mathbf{v}), \quad \mathbf{w} \cdot(\mathbf{v} \times \mathbf{u})}\end{array} $$ (a) Calculate each of these six triple products for the vectors: $$ \mathbf{u}=\langle 0,1,1\rangle, \quad \mathbf{v}=\langle 1,0,1\rangle, \quad \mathbf{w}=\langle 1,1,0\rangle $$ (b) On the basis of your observations in part (a), make a conjecture about the relationships between these six triple products. (c) Prove the conjecture you made in part (b).

\(37-40\) Find \(|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|,|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,\) and \(|\mathbf{u}|-|\mathbf{v}|\) $$ \mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j} $$

Find the direction angles of the given vector, rounded to the nearest degree. $$ \mathbf{i}-2 \mathbf{j}-\mathbf{k} $$

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q .\) $$ \mathbf{F}=-4 \mathbf{i}+20 \mathbf{j} ; \quad P(0,10), Q(5,25) $$

\(37-40\) Find \(|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|,|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,\) and \(|\mathbf{u}|-|\mathbf{v}|\) $$ \mathbf{u}=-2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=\mathbf{i}-2 \mathbf{j} $$

Find the direction angles of the given vector, rounded to the nearest degree. $$ \langle 2,3,-6\rangle $$

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors, and let \(a\) be a scalar. Prove the given property. $$ \mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u} $$

\(37-40\) Find \(|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|,|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,\) and \(|\mathbf{u}|-|\mathbf{v}|\) $$ \mathbf{u}=\langle 10,-1\rangle, \quad \mathbf{v}=\langle- 2,-2\rangle $$

Find the direction angles of the given vector, rounded to the nearest degree. $$ \langle 2,-1,2\rangle $$

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors, and let \(a\) be a scalar. Prove the given property. $$ (a \mathbf{u}) \cdot \mathbf{v}=a(\mathbf{u} \cdot \mathbf{v})=\mathbf{u} \cdot(a \mathbf{v}) $$

\(37-40\) Find \(|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|,|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,\) and \(|\mathbf{u}|-|\mathbf{v}|\) $$ \mathbf{u}=\langle- 6,6\rangle, \quad \mathbf{v}=\langle- 2,-1\rangle $$

Two direction angles of a vector are given. Find the third direction angle, given that it is either obtuse or acute as indicated. (In Exercises 43 and 44, round your answers to the nearest degree.) $$ \alpha=\frac{\pi}{3}, \quad \gamma=\frac{2 \pi}{3} ; \quad \beta \text { is acute } $$

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors, and let \(a\) be a scalar. Prove the given property. $$ (\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{w} $$

\(41-46\) . Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors \(\mathbf{i}\) and \(\mathbf{j}\) . $$ |\mathbf{v}|=40, \quad \theta=30^{\circ} $$

Two direction angles of a vector are given. Find the third direction angle, given that it is either obtuse or acute as indicated. (In Exercises 43 and 44, round your answers to the nearest degree.) $$ \beta=\frac{2 \pi}{3}, \quad \gamma=\frac{\pi}{4} ; \quad \alpha \text { is acute } $$