Question 11.202

Let \({\bf v} = 3 {\bf i} \,{+}\, 4 {\bf j} \,{+}\, 5 {\bf k}\) and \({\bf w} = {\bf i} - {\bf j} + {\bf k}\). Compute \({\bf v} + {\bf w}, 3 {\bf v}, 6 {\bf v} + 8 {\bf w}, -2 {\bf v}, {\bf v \,{\cdot}\, w}, {\bf v} \times {\bf w}\). Interpret each operation geometrically by graphing the vectors.

Question 11.203

Repeat Exercise 1 with \({\bf v} = 2 {\bf j} + {\bf k}\) and \({\bf w} = - {\bf i} - {\bf k}\).

Question 11.204

• (a) Find the equation of the line through \((-1,2,-1)\) in the direction of \({\bf j}\).
• (b) Find the equation of the line passing through \((0,2,-1)\) and \((-3,1,0)\).
• (c) Find the equation for the plane perpendicular to the vector \((-2,1,2)\) and passing through the point \((-1,1,3)\).

Question 11.205

• (a) Find the equation of the line through \((0,1,0)\) in the direction of \(3 {\bf i} + {\bf k}\).
• (b) Find the equation of the line passing through \((0,1,1)\) and \((0,1,0)\).
• (c) Find an equation for the plane perpendicular to the vector \((-1,1,-1)\) and passing through the point \((1,1,1)\).

Question 11.206

Find an equation for the plane containing the points (2, 1, \(-1\)), (3, 0, 2), and (4, \(-3\), 1).

Question 11.207

Find an equation for a line that is parallel to the plane \(2x-3y+5z-10=0\) and passes through the point (\(-1\), 7, 4). (There are lots of them.)

Question 11.208

Compute \({\bf v \,{\cdot}\, w}\) for the following sets of vectors:

• (a) \({\bf v} = - {\bf i} + {\bf j}; {\bf w} = {\bf k}\)
• (b) \({\bf v} = {\bf i}+ 2 {\bf j} - {\bf k} ; {\bf w} = 3 {\bf i} + {\bf j}\)
• (c) \({\bf v} = - 2 {\bf i} - {\bf j} + {\bf k}; {\bf w} = 3 {\bf i} + 2 {\bf j} - 2 {\bf k}\)

Question 11.209

Compute \({\bf v} \times {\bf w}\) for the vectors in Exercise 7. [Only part (b) is solved in the Study Guide.]

Question 11.210

Find the cosine of the angle between the vectors in Exercise 7. [Only part (b) is solved in the Study Guide.]

Question 11.211

Find the area of the parallelogram spanned by the vectors in Exercise 7. [Only part (b) is solved in the Study Guide.]

Question 11.212

Use vector notation to describe the triangle in space whose vertices are the origin and the endpoints of vectors a and b.

Question 11.213

Show that three vectors \({\bf a, b, c}\) lie in the same plane through the origin if and only if there are three scalars \(\alpha , \beta, \gamma\), not all zero, such that \(\alpha {\bf a} + \beta {\bf b} + \gamma {\bf c} ={\bf 0}\).

Question 11.214

For real numbers \(a_1, a_2 ,a_3, b_1, b_2, b_3\), show that \[ (a_1 b_1 + a_2 b_2 + a_3 b_3)^2 \le (a_1^2 +a_2^2 + a_3^2) (b_1^2 + b_2^2 + b_3^2). \]

Question 11.215

Let \({\bf u,v,w}\) be unit vectors that are orthogonal to each other. If \({\bf a}= \alpha {\bf u} + \beta {\bf v} + \gamma {\bf w}\), show that \[ \alpha = {\bf a \,{\cdot}\, u}, \beta = {\bf a \,{\cdot}\, v}, \gamma= {\bf a \,{\cdot}\, w}. \] Interpret the results geometrically.

Question 11.216

Find the products \(AB\) and \(BA\) where \[ A=\left[ \begin{array}{c@{\quad}c@{\quad}c} 1 & 5 & 2 \\ 0 & 2 & 3 \\ 1 & 0 & 2 \\ \end{array} \right] \quad \quad \quad B=\left[ \begin{array}{c@{\quad}c@{\quad}c} 2 & 0 & 1 \\ 1 & 3 & 0 \\ 2 & 4 & 1 \\ \end{array} \right]\!. \]

Question 11.217

Find the products \(AB\) and \(BA\) where \[ A=\left[ \begin{array}{c@{\quad}c@{\quad}c} 2 & 1 & 2 \\ 4 & 0 & 1 \\ 1 & 3 & 0 \\ \end{array} \right] \quad \quad \quad B=\left[ \begin{array}{c@{\quad}c@{\quad}c} 3 & 0 & 5 \\ 1 & 2 & 1 \\ 0 & 3 & 1 \\ \end{array} \right]\!. \]

Question 11.218

Let \({\bf a,b}\) be two vectors in the plane, \({\bf a}=(a_1,a_2), {\bf b} = (b_1, b_2)\), and let \(\lambda\) be a real number. Show that the area of the parallelogram determined by a and \({\bf b} + \lambda {\bf a}\) is the same as that determined by a and b. Sketch. Relate this result to a known property of determinants.

Question 11.219

Find the volume of the parallelepiped determined by the vertices \((0,1,0),(1,1,1),\) \((0,2,0),\) \((3,1,2)\).

Question 11.220

Given nonzero vectors a and b in \({\mathbb R}^3\), show that the vector \({\bf v} = \|{\bf a} \| {\bf b} + \|{\bf b} \| {\bf a}\) bisects the angle between a and b.

Question 11.221

Show that the vectors \(\|\textbf{b}\|\textbf{a} +\|\textbf{a}\|\textbf{b}\) and \(\|\textbf{b}\|\textbf{a} -\|\textbf{a}\|\textbf{b}\) are orthogonal.

Question 11.222

Use the triangle inequality to show that \( \| \textbf{v} - \textbf{w} \| \geq \ \Big| \| \textbf{v} \| - \| \textbf{w} \| \Big| \).

Question 11.223

Use vector methods to prove that the distance from the point \((x_1,y_1)\) to the line \(ax+by\) \(=c\) is \[ \frac{|ax_1 + b y_1 -c|}{\sqrt{a^2 + b^2}}. \]

Question 11.224

Verify that the direction of \({\bf b} \times {\bf c}\) is given by the right-hand rule, by choosing \({\bf b}, {\bf c}\) to be two of the vectors \({\bf i,j}\), and \({\bf k}\).

Question 11.225

• (a) Suppose \({\bf a \,{\cdot}\, b} = {\bf a' \,{\cdot}\, b}\) for all \({\bf b}\). Show that \({\bf a} = {\bf a}'\).
• (b) Suppose \({\bf a} \times {\bf b} = {\bf a}' \times {\bf b}\) for all \({\bf b}\). Is it true that \({\bf a} = {\bf a}'\)?

Question 11.226

• (a) Using vector methods, show that the distance between two nonparallel lines \(l_1\) and \(l_2\) is given by \[ d = \frac{|({\bf v}_2 - {\bf v}_1) \,{\cdot}\, ( {\bf a}_1 \times {\bf a}_2)|}{\|{\bf a}_1 \times {\bf a}_2 \|}, \] where \({\bf v}_1, {\bf v}_2\) are any two points on \(l_1\) and \(l_2\), respectively, and \({\bf a}_1\) and \({\bf a}_2\) are the directions of \(l_1\) and \(l_2\). [HINT: Consider the plane through \(l_2\) that is parallel to \(l_1\). Show that the vector \( ({\bf a}_1 \times {\bf a}_2) / \|{\bf a}_1 \times {\bf a}_2\|\) is a unit normal for this plane; now project \({\bf v}_2 - {\bf v}_1\) onto this normal direction.]
• (b) Find the distance between the line \(l_1\) determined by the points \((-1,-1,1)\) and \((0, 0,0)\) and the line \(l_2\) determined by the points \((0,-2,0)\) and \((2,0,5)\).

Question 11.227

Show that two planes given by the equations \(Ax + By + Cz + D_1=0 \) and \(Ax + By + Cz + D_2 =0\) are parallel, and that the distance between them is \[ \frac{|D_1 - D_2|}{\sqrt{A^2 + B^2 + C^2}}. \]

Question 11.228

• (a) Prove that the area of the triangle in the plane with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is the absolute value of \[ \frac{1}{2}\, \Bigg|\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & 1 & 1\\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{array} \Bigg|. \]
• (b) Find the area of the triangle with vertices \((1,2), (0,1),(-1,1)\).

Question 11.229

Convert the following points from Cartesian to cylindrical and spherical coordinates and plot:

• (a) \((0,3,4)\)
• (b) \((-\sqrt{2},1,0)\)
• (c) \((0,0,0)\)
• (d) \((-1,0,1)\)
• (e) \((-2\sqrt{3}, -2,3)\)

Question 11.230

Convert the following points from cylindrical to Cartesian and spherical coordinates and plot:

• (a) \((1, \pi /4, 1)\)
• (b) \((3, \pi /6, -4)\)
• (c) \((0, \pi /4, 1)\)
• (d) \((2, -\pi /2, 1)\)
• (e) \((-2, -\pi /2, 1)\)

Question 11.231

Convert the following points from spherical to Cartesian and cylindrical coordinates and plot:

• (a) \((1,\pi/2,\pi)\)
• (b) \((2,-\pi/2,\pi/6)\)
• (c) \((0,\pi/8,\pi/35)\)
• (d) \((2,-\pi/2,-\pi)\)
• (e) \((-1,\pi,\pi/6)\)

Question 11.232

Rewrite the equation \(z= x^2 - y^2\) using cylindrical and spherical coordinates.

Question 11.233

Using spherical coordinates, show that \[ \phi = \cos^{-1} \Big( \frac{{\bf u \,{\cdot}\, k}}{\|{\bf u}\|} \Big), \] where \({\bf u} = x {\bf i} + y {\bf j} + z {\bf k}\). Interpret geometrically.

Question 11.234

Verify the Cauchy–Schwarz and triangle inequalities for \[ {\bf x} = (3,2,1,0) \hbox{and} {\bf y} = (1,1,1,2). \]

Question 11.235

Multiply the matrices \[ A = \Bigg[ \begin{array} &3 & 0 & 1 \\ 2 & 0 & 1 \\ 1 & 0 & 1 \end{array} \Bigg] \qquad \hbox{and } \qquad B = \Bigg[ \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & 0 & 1\\ 1 & 1 & 1\\ 0 & 0 & 1 \end{array} \Bigg]. \] Does \(AB =BA\)?

Question 11.236

• (a) Show that for two \(n \times n\) matrices \(A\) and \(B\), and \({\bf x} \in {\mathbb R}^n\), \[ (AB) {\bf x} = A (B {\bf x}). \]
• (b) What does the equality in part (a) imply about the relationship between the composition of the mappings \({\bf x} \mapsto B {\bf x}, {\bf y} \mapsto A {\bf y}\), and matrix multiplication?

Question 11.237

Find the volume of the parallelepiped spanned by the vectors \[ (1,0,1), (1,1,1), \hbox{and} (-3,2,0). \]

Question 11.238

(For students with some knowledge of linear algebra.) Verify that a linear mapping \(T\) of \({\mathbb R}^n\) to \({\mathbb R}^n\) is determined by an \(n \times n\) matrix.

Question 11.239

Find an equation for the plane that contains \((3,-1,2)\) and the line with equation \({\bf v} =(2,-1,0) + t(2,3,0)\).

Question 11.240

The work \(W\) done in moving an object from (0, 0) to (7, 2) subject to a constant force \({\bf F}\) is \(W = {\bf F \,{\cdot}\, r}\), where \({\bf r}\) is the vector with its head at (7, 2) and tail at (0, 0). The units are feet and pounds.

• (a) Suppose the force \({\bf F} = 10\, \cos\, \theta {\bf i} + 10\, \sin \,\theta {\bf j}\). Find \(W\) in terms of \(\theta\).
• (b) Suppose the force \({\bf F}\) has magnitude of 6 lb and makes an angle of \(\pi /6\) rad with the horizontal, pointing right. Find \(W\) in foot-pounds.

Question 11.241

If a particle with mass \(m\) moves with velocity \({\bf v}\), its momentum is \({\bf p} = m { \bf v}\). In a game of marbles, a marble with mass 2 grams (g) is shot with velocity 2 meters per second (m /s), hits two marbles with mass 1 g each, and comes to a dead halt. One of the marbles flies off with a velocity of 3 m/s at an angle of \(45^\circ\) to the incident direction of the larger marble, as in Figure 1.R.1. Assuming that the total momentum before and after the collision is the same (according to the law of conservation of momentum), at what angle and speed does the second marble move?

Figure 11.119: Momentum and marbles.

Question 11.242

Show that for all \(x,y,z\), \[ \Bigg|\begin{array}{@{}c@{\quad}c@{\quad}c@{}} x+2 & y & z \\ z & y+1 & 10 \\ 5 & 5 & 2 \end{array} \Bigg| = - \Bigg| \begin{array}{@{}c@{\quad}c@{\quad}c@{}} y & x+2 & z \\ 1 & z-x-2 & 10-z \\ 5 & 5 & 2 \end{array} \Bigg| . \]

Question 11.243

Show that \[ \Bigg| \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{array} \Bigg| \ne 0 \] if \(x,y\), and \(z\) are all different.

Question 11.244

Show that \[ \Bigg| \begin{array}{@{}r@{\quad}c@{\quad}r@{}} 66 & 628 & 246 \\ 88 & 435 & 24 \\ 2 & -1 & 1 \end{array} \Bigg| = \Bigg| \begin{array}{@{}r@{\quad}c@{\quad}r@{}} 68 & 627 & 247 \\ 86 & 436 & 23 \\ 2 & -1 & 1 \end{array} \Bigg|. \]

Question 11.245

Show that \[ \Bigg| \begin{array}{@{}c@{\qquad}c@{\qquad}c@{}} n & n+1 & n+2 \\ n+3 & n+4 & n+5 \\ n+6 & n+ 7 & n+8 \end{array} \Bigg| \] has the same value no matter what \(n\) is. What is this value?

Question 11.246

Are the following quantities vectors or scalars?

• (a) The current population of Santa Cruz, California
• (b) The torque a cyclist exerts on her bicycle
• (c) The velocity of wind blowing through a weather vane
• (d) The temperature of a pizza in an oven

Question 11.247

Find a \(4 \times 4\) matrix \(C\) such that for every \(4 \times 4\) matrix \(A\) we have \(CA=3A\).

Question 11.248

Let \[ A=\left[ \begin{array}{c@{\quad}c} 1 & 1 \\ 0 & 1 \\ \end{array} \right] \quad \quad \quad B=\left[ \begin{array}{c@{\quad}c} 1 & 0 \\ 2 & 1 \\ \end{array} \right] \]

• (a) Find \(A^{-1}, B^{-1}\), and \((AB)^{-1}\).
• (b) Show that \((AB)^{-1} \neq A^{-1}B^{-1}\) but \((AB)^{-1} = B^{-1}A^{-1}\)

Question 11.249

Suppose \(\left[ \begin{array}{c@{\quad}c} a & b \\ c & d \\ \end{array} \right]\) is invertible and has integer entries.

What conditions must be satisfied for \(A=\left[ \begin{array}{c@{\quad}c} a & b \\ c & d \\ \end{array} \right]^{-1}\) to have integer entries?

Question 11.250

The volume of a tetrahedron with concurrent edges a, b, c is given by \(V = \frac{1}{6} {\bf a} \,{\cdot}\, ( {\bf b} \times {\bf c}).\)

• (a) Express the volume as a determinant.
• (b) Evaluate \(V\) when \({\bf a} = {\bf i} + {\bf j} + {\bf k}, {\bf b} = {\bf i} - {\bf j}+ {\bf k}, {\bf c} = {\bf i} + {\bf j}\).

Question 11.251

A tetrahedron sits in \(xyz\) coordinates with one vertex at \((0,0,0)\), and the three edges concurrent at \((0, 0, 0)\) are coincident with the vectors a, b, c.

• (a) Draw a figure and label the heads of the vectors \({\bf a,b,c}\).
• (b) Find the center of mass of each of the four triangular faces of the tetrahedron if a unit mass is placed at each vertex.

Question 11.252

Show that for any vector \({\bf r}\), the center of mass of a system satisfies \[ \sum_{i=1}^n \,m_i \|{\bf r} - {\bf r}_i\|^2 = \sum_{i=1}^n \,m_i \|{\bf r}_i - {\bf c}\|^2 + m \|{\bf r} - {\bf c} \|^2, \] where \(m = {\sum}_{i=1}^n \,m_i\) is the total mass of the system.

Question 11.253

Parallel to the line \(x=3t +1 , y=16t-2, z =- (t+2)\)

Question 11.254

Orthogonal to the plane \(x-6y +z =12\)

Question 11.255

Parallel to both the planes \(8x + y + z =1\) and \(x-y -z =0\)

Question 11.256

Orthogonal to \({\bf i} + 2 {\bf j} - {\bf k}\) and to k

Question 11.257

Orthogonal to the line \(x = 2t-1, y=-t-1, z=t+2\), and the vector \({\bf i}- {\bf j}\)

Question 11.258

At an angle of \(30^\circ\) to i and making equal angles with j and k