Question 1.174

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 1.175

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

Question 1.176

• (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 1.177

• (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 1.178

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

Question 1.179

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 1.180

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 1.181

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

Question 1.182

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

Question 1.183

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

Question 1.184

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

Question 1.185

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 1.186

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 1.187

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 1.188

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 1.189

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 1.190

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 1.191

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

Question 1.192

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 1.193

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

Question 1.194

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

Question 1.195

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 1.196

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 1.197

• (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 1.198

• (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 1.199

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 1.200

• (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 1.201

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 1.202

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 1.203

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 1.204

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

Question 1.205

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 1.206

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

Question 1.207

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 1.208

• (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 1.209

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

Question 1.210

(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 1.211

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 1.212

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 1.213

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 1.99: Momentum and marbles.

Question 1.214

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 1.215

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 1.216

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 1.217

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 1.218

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 1.219

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

Question 1.220

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 1.221

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 1.222

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 1.223

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 1.224

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 1.225

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

Question 1.226

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

Question 1.227

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

Question 1.228

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

Question 1.229

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

Question 1.230

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