4.2E: Projections and Planes Exercises

Exercises for 1

solutions

2

Compute \(\mathbf{u}\bullet \mathbf{v}\) where:

1. \(\mathbf{u} = \left[ \begin{array}{r} 2\\ -1\\ 3 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} -1\\ 1\\ 1 \end{array} \right]\)
2. \(\mathbf{u} = \left[ \begin{array}{r} 1\\ 2\\ -1 \end{array} \right]\), \(\mathbf{v} = \mathbf{u}\)
3. \(\mathbf{u} = \left[ \begin{array}{r} 1\\ 1\\ -3 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 2\\ -1\\ 1 \end{array} \right]\)
4. \(\mathbf{u} = \left[ \begin{array}{r} 3\\ -1\\ 5 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 6\\ -7\\ -5 \end{array} \right]\)
5. \(\mathbf{u} = \left[ \begin{array}{r} x\\ y\\ z \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} a\\ b\\ c \end{array} \right]\)
6. \(\mathbf{u} = \left[ \begin{array}{r} a\\ b\\ c \end{array} \right]\), \(\mathbf{v} = \mathbf{0}\)

2. \(6\)
3. \(0\)
4. \(0\)

Find the angle between the following pairs of vectors.

1. \(\mathbf{u} = \left[ \begin{array}{r} 1\\ 0\\ 3 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 2\\ 0\\ 1 \end{array} \right]\)
2. \(\mathbf{u} = \left[ \begin{array}{r} 3\\ -1\\ 0 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} -6\\ 2\\ 0 \end{array} \right]\)
3. \(\mathbf{u} = \left[ \begin{array}{r} 7\\ -1\\ 3 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 1\\ 4\\ -1 \end{array} \right]\)
4. \(\mathbf{u} = \left[ \begin{array}{r} 2\\ 1\\ -1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 3\\ 6\\ 3 \end{array} \right]\)
5. \(\mathbf{u} = \left[ \begin{array}{r} 1\\ -1\\ 0 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 0\\ 1\\ 1 \end{array} \right]\)
6. \(\mathbf{u} = \left[ \begin{array}{r} 0\\ 3\\ 4 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 5\sqrt{2}\\ -7\\ -1 \end{array} \right]\)

2. \(\pi\) or \(180^\circ\)
3. \(\frac{\pi}{3}\) or \(60^\circ\)
4. \(\frac{2\pi}{3}\) or \(120^\circ\)

Find all real numbers \(x\) such that:

1. \(\left[ \begin{array}{r} 2\\ -1\\ 3 \end{array} \right]\) and \(\left[ \begin{array}{r} x\\ -2\\ 1 \end{array} \right]\) are orthogonal.
2. \(\left[ \begin{array}{r} 2\\ -1\\ 1 \end{array} \right]\) and \(\left[ \begin{array}{r} 1\\ x\\ 2 \end{array} \right]\) are at an angle of \(\frac{\pi}{3}\).

2. \(1\) or \(-17\)

Find all vectors \(\mathbf{v} = \left[ \begin{array}{r} x\\ y\\ z \end{array} \right]\) orthogonal to both:

1. \(\mathbf{u}_{1} = \left[ \begin{array}{r} -1\\ -3\\ 2 \end{array} \right]\), \(\mathbf{u}_{2} = \left[ \begin{array}{r} 0\\ 1\\ 1 \end{array} \right]\)
2. \(\mathbf{u}_{1} = \left[ \begin{array}{r} 3\\ -1\\ 2 \end{array} \right]\), \(\mathbf{u}_{2} = \left[ \begin{array}{r} 2\\ 0\\ 1 \end{array} \right]\)
3. \(\mathbf{u}_{1} = \left[ \begin{array}{r} 2\\ 0\\ -1 \end{array} \right]\), \(\mathbf{u}_{2} = \left[ \begin{array}{r} -4\\ 0\\ 2 \end{array} \right]\)
4. \(\mathbf{u}_{1} = \left[ \begin{array}{r} 2\\ -1\\ 3 \end{array} \right]\), \(\mathbf{u}_{2} = \left[ \begin{array}{r} 0\\ 0\\ 0 \end{array} \right]\)

2. \(t \left[ \begin{array}{r} -1\\ 1\\ 2 \end{array} \right]\)
3. \(s \left[ \begin{array}{r} 1\\ 2\\ 0 \end{array} \right] + t \left[ \begin{array}{r} 0\\ 3\\ 1 \end{array} \right]\)

Find two orthogonal vectors that are both orthogonal to \(\mathbf{v} = \left[ \begin{array}{r} 1\\ 2\\ 0 \end{array} \right]\).

Consider the triangle with vertices \(P(2, 0, -3)\), \(Q(5, -2, 1)\), and \(R(7, 5, 3)\).

1. Show that it is a right-angled triangle.
2. Find the lengths of the three sides and verify the Pythagorean theorem.

2. \(29 + 57 = 86\)

Show that the triangle with vertices \(A(4, -7, 9)\), \(B(6, 4, 4)\), and \(C(7, 10, -6)\) is not a right-angled triangle.

Find the three internal angles of the triangle with vertices:

1. \(A(3, 1, -2)\), \(B(3, 0, -1)\), and \(C(5, 2, -1)\)
2. \(A(3, 1, -2)\), \(B(5, 2, -1)\), and \(C(4, 3, -3)\)

2. \(A = B = C = \frac{\pi}{3}\) or \(60^\circ\)

Show that the line through \(P_{0}(3, 1, 4)\) and \(P_{1}(2, 1, 3)\) is perpendicular to the line through \(P_{2}(1, -1, 2)\) and \(P_{3}(0, 5, 3)\).

In each case, compute the projection of \(\mathbf{u}\) on \(\mathbf{v}\).

1. \(\mathbf{u} = \left[ \begin{array}{r} 5\\ 7\\ 1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 2\\ -1\\ 3 \end{array} \right]\)
2. \(\mathbf{u} = \left[ \begin{array}{r} 3\\ -2\\ 1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 4\\ 1\\ 1 \end{array} \right]\)
3. \(\mathbf{u} = \left[ \begin{array}{r} 1\\ -1\\ 2 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 3\\ -1\\ 1 \end{array} \right]\)
4. \(\mathbf{u} = \left[ \begin{array}{r} 3\\ -2\\ -1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} -6\\ 4\\ 2 \end{array} \right]\)

2. \(\frac{11}{18}\mathbf{v}\)
3. \(-\frac{1}{2}\mathbf{v}\)

In each case, write \(\mathbf{u} = \mathbf{u}_{1} + \mathbf{u}_{2}\), where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v}\).

1. \(\mathbf{u} = \left[ \begin{array}{r} 2\\ -1\\ 1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 1\\ -1\\ 3 \end{array} \right]\)
2. \(\mathbf{u} = \left[ \begin{array}{r} 3\\ 1\\ 0 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} -2\\ 1\\ 4 \end{array} \right]\)
3. \(\mathbf{u} = \left[ \begin{array}{r} 2\\ -1\\ 0 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 3\\ 1\\ -1 \end{array} \right]\)
4. \(\mathbf{u} = \left[ \begin{array}{r} 3\\ -2\\ 1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} -6\\ 4\\ -1 \end{array} \right]\)

2. \(\frac{5}{21}\left[ \begin{array}{r} 2\\ -1\\ -4 \end{array} \right] + \frac{1}{21}\left[ \begin{array}{r} 53\\ 26\\ 20 \end{array} \right]\)
3. \(\frac{27}{53}\left[ \begin{array}{r} 6\\ -4\\ 1 \end{array} \right] + \frac{1}{53}\left[ \begin{array}{r} -3\\ 2\\ 26 \end{array} \right]\)

Calculate the distance from the point \(P\) to the line in each case and find the point \(Q\) on the line closest to \(P\).

1. \(P(3,2-1) \quad\)
   line: \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ 1\\ 3 \end{array} \right] +t \left[ \begin{array}{r} 3\\ -1\\ -2 \end{array} \right]\)

2. \(P(1,-1,3) \quad\)
   line: \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 0\\ -1 \end{array} \right] +t \left[ \begin{array}{r} 3\\ 1\\ 4 \end{array} \right]\)

2. \(\frac{1}{26}\sqrt{5642}\), \(Q(\frac{71}{26}, \frac{15}{26}, \frac{34}{26})\)

Compute \(\mathbf{u} \times \mathbf{v}\) where:

1. \(\mathbf{u} = \left[ \begin{array}{r} 1\\ 2\\ 3 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 1\\ 1\\ 2 \end{array} \right]\)
2. \(\mathbf{u} = \left[ \begin{array}{r} 3\\ -1\\ 0 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} -6\\ 2\\ 0 \end{array} \right]\)
3. \(\mathbf{u} = \left[ \begin{array}{r} 3\\ -2\\ 1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 1\\ 1\\ -1 \end{array} \right]\)
4. \(\mathbf{u} = \left[ \begin{array}{r} 2\\ 0\\ -1 \end{array} \right]\), \(\mathbf{v} = \left[ \begin{array}{r} 1\\ 4\\ 7 \end{array} \right]\)

2. \(\left[ \begin{array}{r} 0\\ 0\\ 0 \end{array} \right]\)
3. \(\left[ \begin{array}{r} 4\\ -15\\ 8 \end{array} \right]\)

Find an equation of each of the following planes.

1. Passing through \(A(2, 1, 3)\), \(B(3, -1, 5)\), and \(C(1, 2, -3)\).
2. Passing through \(A(1, -1, 6)\), \(B(0, 0, 1)\), and \(C(4, 7, -11)\).
3. Passing through \(P(2, -3, 5)\) and parallel to the plane with equation \(3x - 2y - z = 0\).
4. Passing through \(P(3, 0, -1)\) and parallel to the plane with equation \(2x - y + z = 3\).
5. Containing \(P(3, 0, -1)\) and the line \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 0\\ 0\\ 2 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 0\\ 1 \end{array} \right].\)
6. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 3\\ -1\\ 2 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 0\\ -1 \end{array} \right].\)
7. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ -1\\ 2 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 1\\ 1 \end{array} \right]\) and \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 0\\ 0\\ 2 \end{array} \right] +t \left[ \begin{array}{r} 1\\ -1\\ 0 \end{array} \right]\).
8. Containing the lines \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 3\\ 1\\ 0 \end{array} \right] +t \left[ \begin{array}{r} 1\\ -1\\ 3 \end{array} \right]\) and \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 0\\ -2\\ 5 \end{array} \right] +t \left[ \begin{array}{r} 2\\ 1\\ -1 \end{array} \right]\).
9. Each point of which is equidistant from \(P(2, -1, 3)\) and \(Q(1, 1, -1)\).
10. Each point of which is equidistant from \(P(0, 1, -1)\) and \(Q(2, -1, -3)\).

2. \(-23x + 32y + 11z = 11\)
3. \(2x - y + z = 5\)
4. \(2x + 3y + 2z = 7\)
5. \(2x - 7y - 3z = -1\)
6. \(x - y - z = 3\)

In each case, find a vector equation of the line.

1. Passing through \(P(3, -1, 4)\) and perpendicular to the plane \(3x - 2y - z = 0\).
2. Passing through \(P(2, -1, 3)\) and perpendicular to the plane \(2x + y = 1\).
3. Passing through \(P(0, 0, 0)\) and perpendicular to the lines \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 1\\ 0 \end{array} \right] +t \left[ \begin{array}{r} 2\\ 0\\ -1 \end{array} \right]\) and \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ 1\\ -3 \end{array} \right] +t \left[ \begin{array}{r} 1\\ -1\\ 5 \end{array} \right]\).
4. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ 0\\ 1 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 1\\ -2 \end{array} \right]\) and
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 5\\ 5\\ -2 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 2\\ -3 \end{array} \right]\).
5. Passing through \(P(2, 1, -1)\), intersecting the line \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 2\\ -1 \end{array} \right] +t \left[ \begin{array}{r} 3\\ 0\\ 1 \end{array} \right]\), and perpendicular to that line.
6. Passing through \(P(1, 1, 2)\), intersecting the line \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ 1\\ 0 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 1\\ 1 \end{array} \right]\), and perpendicular to that line.

2. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ -1\\ 3 \end{array} \right] +t \left[ \begin{array}{r} 2\\ 1\\ 0 \end{array} \right]\)
3. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 1\\ -1 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 1\\ 1 \end{array} \right]\)
4. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 1\\ 2 \end{array} \right] +t \left[ \begin{array}{r} 4\\ 1\\ -5 \end{array} \right]\)

In each case, find the shortest distance from the point \(P\) to the plane and find the point \(Q\) on the plane closest to \(P\).

1. \(P(2, 3, 0)\); plane with equation \(5x + y + z = 1\).
2. \(P(3, 1, -1)\); plane with equation \(2x + y - z = 6\).

2. \(\frac{\sqrt{6}}{3}\), \(Q(\frac{7}{3}, \frac{2}{3}, \frac{-2}{3})\)

1. Does the line through \(P(1, 2, -3)\) with direction vector \(\mathbf{d} = \left[ \begin{array}{r} 1\\ 2\\ -3 \end{array} \right]\) lie in the plane \(2x - y - z = 3\)? Explain.
2. Does the plane through \(P(4, 0, 5)\), \(Q(2, 2, 1)\), and \(R(1, -1, 2)\) pass through the origin? Explain.

2. Yes. The equation is \(5x -3y - 4z = 0\).

Show that every plane containing \(P(1, 2, -1)\) and \(Q(2, 0, 1)\) must also contain \(R(-1, 6, -5)\).

Find the equations of the line of intersection of the following planes.

1. \(2x -3y + 2z = 5\) and \(x + 2y - z = 4\).
2. \(3x + y -2z = 1\) and \(x + y + z = 5\).

2. \((-2, 7, 0) + t(3, -5, 2)\)

In each case, find all points of intersection of the given plane and the line \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ -2\\ 3 \end{array} \right] +t \left[ \begin{array}{r} 2\\ 5\\ -1 \end{array} \right]\).

\(x -3y + 2z = 4\) \(2x - y - z = 5\) \(3x - y + z = 8\) \(-x -4y -3z = 6\)

2. None
3. \(P(\frac{13}{19}, \frac{-78}{19}, \frac{65}{19})\)

Find the equation of all planes:

1. Perpendicular to the line
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ -1\\ 3 \end{array} \right] +t \left[ \begin{array}{r} 2\\ 1\\ 3 \end{array} \right]\).

2. Perpendicular to the line
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 0\\ -1 \end{array} \right] +t \left[ \begin{array}{r} 3\\ 0\\ 2 \end{array} \right]\).

3. Containing the origin.
4. Containing \(P(3, 2, -4)\).
5. Containing \(P(1, 1, -1)\) and \(Q(0, 1, 1)\).
6. Containing \(P(2, -1, 1)\) and \(Q(1, 0, 0)\).

7. Containing the line
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ 1\\ 0 \end{array} \right] +t \left[ \begin{array}{r} 1\\ -1\\ 0 \end{array} \right]\).

8. Containing the line
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 3\\ 0\\ 2 \end{array} \right] +t \left[ \begin{array}{r} 1\\ -2\\ -1 \end{array} \right]\).

2. \(3x + 2z = d\), \(d\) arbitrary
3. \(a(x - 3) + b(y - 2) + c(z + 4) = 0\); \(a\), \(b\), and \(c\) not all zero
4. \(ax + by + (b - a)z = a\); \(a\) and \(b\) not both zero
5. \(ax + by + (a - 2b)z = 5a - 4b\); \(a\) and \(b\) not both zero

If a plane contains two distinct points \(P_{1}\) and \(P_{2}\), show that it contains every point on the line through \(P_{1}\) and \(P_{2}\).

Find the shortest distance between the following pairs of parallel lines.

1. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ -1\\ 3 \end{array} \right] +t \left[ \begin{array}{r} 1\\ -1\\ 4 \end{array} \right];\)
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 0\\ 1 \end{array} \right] +t \left[ \begin{array}{r} 1\\ -1\\ 4 \end{array} \right]\)

2. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 3\\ 0\\ 2 \end{array} \right] +t \left[ \begin{array}{r} 3\\ 1\\ 0 \end{array} \right];\)
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} -1\\ 2\\ 2 \end{array} \right] +t \left[ \begin{array}{r} 3\\ 1\\ 0 \end{array} \right]\)

2. \(\sqrt{10}\)

Find the shortest distance between the following pairs of nonparallel lines and find the points on the lines that are closest together.

1. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 3\\ 0\\ 1 \end{array} \right] +s \left[ \begin{array}{r} 2\\ 1\\ -3 \end{array} \right];\)
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 1\\ -1 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 0\\ 1 \end{array} \right]\)

2. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ -1\\ 0 \end{array} \right] +s \left[ \begin{array}{r} 1\\ 1\\ 1 \end{array} \right];\)
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 2\\ -1\\ 3 \end{array} \right] +t \left[ \begin{array}{r} 3\\ 1\\ 0 \end{array} \right]\)

3. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 3\\ 1\\ -1 \end{array} \right] +s \left[ \begin{array}{r} 1\\ 1\\ -1 \end{array} \right];\)
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 2\\ 0 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 0\\ 2 \end{array} \right]\)

4. \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 1\\ 2\\ 3 \end{array} \right] +s \left[ \begin{array}{r} 2\\ 0\\ -1 \end{array} \right];\)
   \(\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 3\\ -1\\ 0 \end{array} \right] +t \left[ \begin{array}{r} 1\\ 1\\ 0 \end{array} \right]\)

2. \(\frac{\sqrt{14}}{2}\), \(A(3, 1, 2)\), \(B(\frac{7}{2}, -\frac{1}{2}, 3)\)
3. \(\frac{\sqrt{6}}{6}\), \(A(\frac{19}{3}, 2, \frac{1}{3})\), \(B(\frac{37}{6}, \frac{13}{6}, 0)\)

Show that two lines in the plane with slopes \(m_{1}\) and \(m_{2}\) are perpendicular if and only if
\(m_{1}m_{2} = -1\). [Hint: Example [exa:011343].]

1. Show that, of the four diagonals of a cube, no pair is perpendicular.
2. Show that each diagonal is perpendicular to the face diagonals it does not meet.

2. Consider the diagonal \(\mathbf{d} = \left[ \begin{array}{r} a\\ a\\ a \end{array} \right]\) The six face diagonals in question are \(\pm\left[ \begin{array}{r} a\\ 0\\ -a \end{array} \right]\), \(\pm\left[ \begin{array}{r} 0\\ a\\ -a \end{array} \right]\), \(\pm\left[ \begin{array}{r} a\\ -a\\ 0 \end{array} \right]\). All of these are orthogonal to \(\mathbf{d}\). The result works for the other diagonals by symmetry.

Given a rectangular solid with sides of lengths \(1\), \(1\), and \(\sqrt{2}\), find the angle between a diagonal and one of the longest sides.

Consider a rectangular solid with sides of lengths \(a\), \(b\), and \(c\). Show that it has two orthogonal diagonals if and only if the sum of two of \(a^{2}\), \(b^{2}\), and \(c^{2}\) equals the third.

The four diagonals are \((a, b, c)\), \((-a, b, c)\), \((a, -b, c)\) and \((a, b, -c)\) or their negatives. The dot products are \(\pm(-a^{2} + b^{2} + c^{2})\), \(\pm(a^{2} - b^{2} + c^{2})\), and \(\pm(a^{2} + b^{2} - c^{2})\).

Let \(A\), \(B\), and \(C(2, -1, 1)\) be the vertices of a triangle where \(\longvect{AB}\) is parallel to \(\left[ \begin{array}{r} 1\\ -1\\ 1 \end{array} \right]\), \(\longvect{AC}\) is parallel to \(\left[ \begin{array}{r} 2\\ 0\\ -1 \end{array} \right]\), and angle \(C = 90^\circ\). Find the equation of the line through \(B\) and \(C\).

If the diagonals of a parallelogram have equal length, show that the parallelogram is a rectangle.

Given \(\mathbf{v} = \left[ \begin{array}{r} x\\ y\\ z \end{array} \right]\) in component form, show that the projections of \(\mathbf{v}\) on \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are \(x\mathbf{i}\), \(y\mathbf{j}\), and \(z\mathbf{k}\), respectively.

1. Can \(\mathbf{u}\bullet \mathbf{v} = -7\) if \(\|\mathbf{u}\| = 3\) and \(\|\mathbf{v}\| = 2\)? Defend your answer.
2. Find \(\mathbf{u}\bullet \mathbf{v}\) if \(\mathbf{u} = \left[ \begin{array}{r} 2\\ -1\\ 2 \end{array} \right]\), \(\| \mathbf{v} \| = 6\), and the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(\frac{2\pi}{3}\).

Show \((\mathbf{u} + \mathbf{v})\bullet (\mathbf{u} - \mathbf{v}) = \|\mathbf{u}\|^{2} - \|\mathbf{v}\|^{2}\) for any vectors \(\mathbf{u}\) and \(\mathbf{v}\).

1. Show \(\|\mathbf{u} + \mathbf{v}\|^{2} + \|\mathbf{u} - \mathbf{v}\|^{2} = 2(\|\mathbf{u}\|^{2} + \|\mathbf{v}\|^{2})\) for any vectors \(\mathbf{u}\) and \(\mathbf{v}\).
2. What does this say about parallelograms?

2. The sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the four sides.

Show that if the diagonals of a parallelogram are perpendicular, it is necessarily a rhombus. [Hint: Example [exa:011899].]

Let \(A\) and \(B\) be the end points of a diameter of a circle (see the diagram). If \(C\) is any point on the circle, show that \(AC\) and \(BC\) are perpendicular. [Hint: Express \(\longvect{AB}\bullet (\longvect{AB} \times \longvect{AC}) = 0\) and \(\longvect{BC}\) in terms of \(\mathbf{u} = \longvect{OA}\) and \(\mathbf{v} = \longvect{OC}\), where \(O\) is the centre.]

Show that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, if and only if \(\|\mathbf{u} + \mathbf{v}\|^{2} = \|\mathbf{u}\|^{2} + \|\mathbf{v}\|^{2}\).

Let \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) be pairwise orthogonal vectors.

1. Show that \(\|\mathbf{u} + \mathbf{v} + \mathbf{w}\|^{2} = \|\mathbf{u}\|^{2} + \|\mathbf{v}\|^{2} + \|\mathbf{w}\|^{2}\).
2. If \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are all the same length, show that they all make the same angle with \(\mathbf{u} + \mathbf{v} + \mathbf{w}\).

2. The angle \(\theta\) between \(\mathbf{u}\) and \((\mathbf{u} + \mathbf{v} + \mathbf{w})\) is given by \(\cos\theta = \frac{\mathbf{u}\bullet (\mathbf{u} + \mathbf{v} + \mathbf{w})}{\| \mathbf{u} \| \| \mathbf{u} + \mathbf{v} + \mathbf{w} \|} = \frac{\| \mathbf{u} \|}{\sqrt{\| \mathbf{u} \|^2 + \| \mathbf{v} \|^2 + \| \mathbf{w} \|^2}} = \frac{1}{\sqrt{3}}\) because \(\| \mathbf{u} \| = \| \mathbf{v} \| = \| \mathbf{w} \|\). Similar remarks apply to the other angles.

1. Show that \(\mathbf{n} = \left[ \begin{array}{r} a\\ b \end{array} \right]\) is orthogonal to every vector along the line \(ax + by + c = 0\).
2. [Hint: If \(P_{1}\) is on the line, project \(\mathbf{u} = \longvect{P_{1}P}_{0}\) on \(\mathbf{n}\).]

2. Let \(\mathbf{p}_{0}\), \(\mathbf{p}_{1}\) be the vectors of \(P_{0}\), \(P_{1}\), so \(\mathbf{u} = \mathbf{p}_{0} - \mathbf{p}_{1}\). Then \(\mathbf{u} \cdot \mathbf{n} = \mathbf{p}_{0} \cdot \mathbf{n}\) – \(\mathbf{p}_{1} \cdot \mathbf{n} = (ax_{0} + by_{0}) - (ax_{1} + by_{1}) = ax_{0} + by_{0} + c\). Hence the distance is

   \[\left\| \left( \frac{\mathbf{u}\bullet \mathbf{n}}{\| \mathbf{n} \|^2}\right)\mathbf{n} \right\| = \frac{|\mathbf{u}\bullet \mathbf{n}|}{\| \mathbf{n} \|} \nonumber \]

Assume \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero vectors that are not parallel. Show that \(\mathbf{w} = \|\mathbf{u}\|\mathbf{v} + \|\mathbf{v}\|\mathbf{u}\) is a nonzero vector that bisects the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

Let \(\alpha\), \(\beta\), and \(\gamma\) be the angles a vector \(\mathbf{v} \neq \mathbf{0}\) makes with the positive \(x\), \(y\), and \(z\) axes, respectively. Then \(\cos \alpha\), \(\cos \beta\), and \(\cos \gamma\) are called the direction cosines of the vector \(\mathbf{v}\).

1. If \(\mathbf{v} = \left[ \begin{array}{r} a\\ b\\ c \end{array} \right]\), show that \(\cos\alpha = \frac{a}{\| \mathbf{v} \|}\), \(\cos\beta = \frac{b}{\| \mathbf{v} \|}\), and \(\cos\gamma= \frac{c}{\| \mathbf{v} \|}\).
2. Show that \(\cos^{2} \alpha + \cos^2 \beta + \cos^2 \gamma = 1\).

2. This follows from (a) because \(\|\mathbf{v}\|^{2} = a^{2} + b^{2} + c^{2}\).

Let \(\mathbf{v} \neq \mathbf{0}\) be any nonzero vector and suppose that a vector \(\mathbf{u}\) can be written as \(\mathbf{u} = \mathbf{p} + \mathbf{q}\), where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{q}\) is orthogonal to \(\mathbf{v}\). Show that \(\mathbf{p}\) must equal the projection of \(\mathbf{u}\) on \(\mathbf{v}\). [Hint: Argue as in the proof of Theorem [thm:011958].]

Let \(\mathbf{v} \neq \mathbf{0}\) be a nonzero vector and let \(a \neq 0\) be a scalar. If \(\mathbf{u}\) is any vector, show that the projection of \(\mathbf{u}\) on \(\mathbf{v}\) equals the projection of \(\mathbf{u}\) on \(a\mathbf{v}\).

1. Show that the Cauchy-Schwarz inequality \(|\mathbf{u}\bullet \mathbf{v}| \leq \|\mathbf{u}\|\|\mathbf{v}\|\) holds for all vectors \(\mathbf{u}\) and \(\mathbf{v}\). [Hint: \(|\cos \theta| \leq 1\) for all angles \(\theta\).]
2. [Hint: When is \(\cos \theta = \pm 1\)?]
3. holds for all numbers \(x_{1}\), \(x_{2}\), \(y_{1}\), \(y_{2}\), \(z_{1}\), and \(z_{2}\).
4. Show that \(|xy + yz + zx| \leq x^{2} + y^{2} + z^{2}\) for all \(x\), \(y\), and \(z\).
5. Show that \((x + y + z)^{2} \leq 3(x^{2} + y^{2} + z^{2})\) holds for all \(x\), \(y\), and \(z\).

4. Take \(\left[ \begin{array}{c} x_{1}\\ y_{1}\\ z_{1} \end{array} \right] = \left[ \begin{array}{c} x\\ y\\ z \end{array} \right]\) and \(\left[ \begin{array}{c} x_{2}\\ y_{2}\\ z_{2} \end{array} \right] = \left[ \begin{array}{c} y\\ z\\ x \end{array} \right]\) in (c).

Prove that the triangle inequality \(\| \mathbf{u} + \mathbf{v} \| \leq \|\mathbf{u}\| + \|\mathbf{v}\|\) holds for all vectors \(\mathbf{u}\) and \(\mathbf{v}\). [Hint: Consider the triangle with \(\mathbf{u}\) and \(\mathbf{v}\) as two sides.]