Chapter 1
Section 1.1
2. (a) $ (4, 5, 0) $
(b) $ (12, 5, 4) $
(c) $ (-12, 8, -{8}) $
(d) $ (-11, 7, -5) $
5. (a) $ \sqrt{14} $
(b) $ \sqrt{118} $
(c) $ 5\sqrt{14} $
(d) $ 3\sqrt{6} $
6. (a) $ \sqrt{17} $
(b) $ \sqrt{22} $
(c) $ 2\sqrt{3} $
(d) $ \sqrt{66} $
(e) $ \sqrt{19} $
11. No
12. (a) $ ||\mathbf{x}|| = \sqrt{5} $, Direction: $ \displaystyle{||\mathbf{u}|| = \frac{1}{\sqrt{5}}(2, 1)}$
(b) $ ||\mathbf{z}|| = \sqrt{3} $, Direction: $ \displaystyle{||\mathbf{u}|| = \frac{1}{\sqrt{3}}(1, 1, -1)}$
(c) $ ||\mathbf{x}|| = \sqrt{14} $, Direction: $ \displaystyle{||\mathbf{u}|| = \frac{1}{\sqrt{14}}(-1, 2, 3)}$
(d) $ ||\mathbf{w}|| = \sqrt{15} $, Direction: $ \displaystyle{||\mathbf{u}|| = \frac{1}{\sqrt{15}}(1, -1, 2, -3)}$
15. (a) $ a = \dfrac{3}{2} $, $ b = -\dfrac{1}{2} $; Yes, $ a $ and $ b $ are unique.
(b) $ a = \dfrac{x + y}{2} $, $ b = \dfrac{y - x}{2} $
Section 1.2
1. (a) $ -16 $
(b) $ -32 $
(c) $ -58 $
(d) $ 5 $
3. (a) $ -18 $
(b) $ -36 $
(c) $ -40 $
(d) $ -126 $
4. (a) $ 0.6435 $ radians, or $ 36.87^\circ $
(c) $ 1.9106$ radians, or $ 109.47^\circ$
(e) $ 0.6435 $ radians, or $ 36.87^\circ $
5. The angle at vertex $ (-2, 1) $ is $ 0.3218 $ radians, at vertex $ (1, 2) $ is $ 2.0344 $ radians, and at vertex $ (2, 1) $ is $ \frac{\pi}{4} $ radians.
7. (a) $ 2.1588 $ radians; $ 0.5880 $ radians
(b) $ 1.9913 $ radians; $ 0.6155 $ radians; $ 1.1503 $ radians
(c) $ 1.0282 $ radians; $ 0.6847 $ radians; $ 1.3096 $ radians; $ 1.8320 $ radians
(d) $ 1.4355 $ radians; $ 1.2977 $ radians; $ 1.1543 $ radians; $ 1.011 $ radians; $ 0.8309 $ radians
8. (a) Coordinate: $ -\dfrac{4}{\sqrt{17}} $; Projection: $ \displaystyle{\left(-\frac{16}{17},-\frac{4}{17}\right)}$
(b) Coordinate: $ \dfrac{3}{\sqrt{11}} $; Projection: $ \displaystyle{\left(-\frac{3}{11},\frac{9}{11},\frac{3}{11}\right)}$
(c) Coordinate: $ \dfrac{5}{\sqrt{38}} $; Projection: $ \displaystyle{\left(\frac{5}{38},-\frac{5}{38},\frac{15}{19}\right)}$
(d) Coordinate: $ \dfrac{5}{3\sqrt{2}} $; Projection: $ \displaystyle{\left(\frac{5}{9},-\frac{5}{18},\frac{5}{9},\frac{5}{6}\right)}$
11. $ \displaystyle{\mathbf{x} = \left(\frac{5}{7},\frac{15}{14},\frac{5}{14}\right) + \left(\frac{2}{7}, \frac{13}{14},-\frac{47}{14}\right)}$
Section 1.3
1. (a) $ \mathbf{x} \times \mathbf{y} = (1, 3, 7) $
(b) $ \mathbf{x} \times \mathbf{y} = (-2, 16, -5) $
(c) $ \mathbf{x} \times \mathbf{y} = (36, -12, 0) $
(d) $ \mathbf{x} \times \mathbf{y} = (-6, 2, -14) $
3. $ 11 $
5. $ 3 $
7. $ \dfrac{9}{2} $
9. $ 42 $
13. For example, $ \mathbf{e}_2 \times \left(\mathbf{e}_2 \times \mathbf{e}_3\right) = -\mathbf{e}_3 $, whereas $ \left(\mathbf{e}_2 \times \mathbf{e}_2\right) \times \mathbf{e}_3 = \mathbf{0} $.
Section 1.4
1. Vector equation: $ \mathbf{y} = t(1, -2) + (2, 3) = (t + 2, -2t + 3) $
Parametric equations: \[ \begin{align} x &= t +2 \\ y &= -2t + 3 \end{align} \]
3. (a) Vector equation: $ \mathbf{y} = t(5,5) + (-1, -3) = (5t - 1, 5t - 3) $
Parametric equations: \[ \begin{align} x &= 5t -1 \\ y &= 5t -3 \end{align} \]
(b) Vector equation:
$ \mathbf{y} = t(3, 1, -2) + (2, 1, 3) = (3t+2, t+1, -2t+3) $
Parametric equations: \[ \begin{align} x &= 3t + 2 \\ y &= t + 1 \\ z &= -2t + 3 \end{align} \]
(c) Vector equation:
$ \mathbf{y} = t(1, 2, -3, 3) + (3, 2, 1, 4) = (t+3, 2t+2,-3t+1,3t+4) $
Parametric equations: \[ \begin{align} w &= t + 3 \\ x &= 2t + 2 \\ y &= -3t + 1 \\ z &= 3t + 4 \end{align} \]
(d) Vector equation:
$ \mathbf{y} = t(3, -1, -2) + (4, -3, 2) = (3t+4, -t-3, -2t+2) $
Parametric equations: \[ \begin{align} x &= 3t + 4 \\ y &= -t - 3 \\ z &= -2t + 2 \end{align} \]
5. $ \dfrac{1085}{7} $
7. $ \dfrac{4697}{14} $
9. Vector equation:
$ \mathbf{y} = t(-3, 6, -2, -3) + s(0, 2, -2, 2) + (2, 3, 4, -1) $
Parametic equations: \[ \begin{align} w &= -3t + 2 \\ x &= 6t + 2s + 3 \\ y &= -2t - 2s + 4 \\ z &= -3t + 2s - 1 \end{align} \]
11. $ 3 $
13. $ \mathbf{n} = (1, 0) $
Normal equation: $ (1, 0) \cdot (x-2, y) = 0 $, or $ x = 2 $
15. $ \mathbf{n} = (1, 4) $
Normal equation: $ (1, 4) \cdot (x-3, y-2) = 0 $, or $ x + 4y = 11 $
16. $ \mathbf{n} = (11, 8, 7) $
Normal equation:
$ (11, 8, 7) \cdot (x-1, y-2, z+1) = 0 $, or $ 11x + 8y + 7z = 20 $
17. $ \dfrac{4}{\sqrt{5}} $
19. $ \dfrac{3}{\sqrt{23}} $
21. $ 0.7017 $ radians
23. $ 2x - y = 3 $ is the equation of one such plane.
27. $ y = 2t - \frac{2}{3} $, $ z = -s - t + \frac{11}{3} $
Section 1.5
2. (a) Dimension of the domain space $ = 2 $; dimension of the range space $ = 3 $; $ f $ is linear
(b) Dimension of the domain space $ = 2 $; dimension of the range space $ = 2 $, $ f $ is neither linear nor affine
(c) Dimension of the domain space $ = 3$; dimension of the range space $ = 3$; $ f$ is linear
(d) Dimension of the domain space $ = 3 $; dimension of the range space $ = 2 $; $ f $ is linear
(e) Dimension of the domain space $ = 3 $; dimension of the range space $ = 4 $; $ f $ is affine
(f) Dimension of the domain space $ = 2 $; dimension of the range space $ = 1 $; $ f $ is affine
(g) Dimension of the domain space $ = 1 $; dimension of the range space $ = 2 $; $ f $ is linear
(h) Dimension of the domain space $ = 4 $; dimension of the range space $ = 2 $; $ f $ is linear
(i) Dimension of the domain space $ = 2 $; dimension of the range space $ = 2 $; $ f $ is neither linear nor affine
(j) Dimension of the domain space $ = 2 $; dimension of the range space $ = 3 $; $ f $ is neither linear nor affine
3. (a) $ \displaystyle{M = \begin{bmatrix}1 & \phantom{-}1\\ 2 & -3\end{bmatrix}} $
(b) $ \displaystyle{M = \begin{bmatrix}2 & 1 & -1 & \phantom{-}3\\ 1 & 2 & \phantom{-}0 & -3 \end{bmatrix}}$
(c) $ \displaystyle{M = \begin{bmatrix}3 \\ 1 \\ 4\end{bmatrix}}$
(d) $ \displaystyle{M = \begin{bmatrix}-5\end{bmatrix}}$
(e) $ \displaystyle{M = \begin{bmatrix}4 & -3 & 2\end{bmatrix}}$
(f) $ \displaystyle{M = \begin{bmatrix}1 & \phantom{-}1 & 1\\ 3 & -1 & 0\\ 0 & \phantom{-}1 & 2\end{bmatrix}}$
(g) $ \displaystyle{M = \begin{bmatrix}2 & \phantom{-}0\\ 0 & \phantom{-}3\\ 1 & \phantom{-}1\\ 1 & -1\\ 2 & -3\end{bmatrix}}$
(h) $ \displaystyle{M = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}}$
(i) $ \displaystyle{M = \begin{bmatrix}2 & 1 & -1 & \phantom{-}3\\ 1 & 2 & \phantom{-}0 & -3\end{bmatrix}}$
5. (a) $ \displaystyle{\begin{bmatrix}-4 \\ \phantom{-}4\end{bmatrix}}$
(b) $ \displaystyle{\begin{bmatrix}-5\\ 11\\ -4\end{bmatrix}}$
(c) $ \displaystyle{\begin{bmatrix}3\end{bmatrix}}$
(d) $ \displaystyle{\begin{bmatrix}2 \\ 10\\ 3\end{bmatrix}}$
7. $ \displaystyle{M = \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}}$
8. $ \displaystyle{M = \begin{bmatrix}\phantom{-}0 & -1\\ -1 & \phantom{-}0\end{bmatrix}}$
10. $ \displaystyle{M = \begin{bmatrix}\phantom{-}\cos(\theta) & \phantom{-}\sin(\theta)\\ -\sin(\theta) & \phantom{-}\cos(\theta)\end{bmatrix}}$
Section 1.6
1. (a) $ \displaystyle{\begin{bmatrix}\phantom{-}6 & \phantom{-}9\\ -6 & \phantom{-}3\\ 12 & -3\end{bmatrix}}$
(b) $ \displaystyle{\begin{bmatrix}-1 & 5\\ -3 & 1\\ \phantom{-}2 & 4\end{bmatrix}}$
(c) $ \displaystyle{\begin{bmatrix}\phantom{-}7 & \phantom{-}4\\ -3 & \phantom{-}2\\ 10 & -7\end{bmatrix}}$
(d) $ \displaystyle{\begin{bmatrix}-14 & 18\\ -10 & 2\\ -4 & 28\end{bmatrix}}$
2. (a) $ \displaystyle{\begin{bmatrix}12\\ 1\end{bmatrix}}$
(b) $ \displaystyle{\begin{bmatrix}-4 & 14\\ \phantom{-}9 & -4\end{bmatrix}}$
(c) $ \displaystyle{\begin{bmatrix}12 & 13 & -4\\ -7 & -7 & \phantom{-}9\end{bmatrix}}$
(d) $ \displaystyle{\begin{bmatrix}2 & 19\end{bmatrix}}$
3. (a) $ \displaystyle{\begin{bmatrix}\phantom{-}6 & 9 & \phantom{-}0\\ -3 & 3 & \phantom{-}6\\ \phantom{-}3 & 6 & -3\end{bmatrix}}$
(b) $ \displaystyle{\begin{bmatrix}4 & 7 & -3\\ 0 & 2 & \phantom{-}3\\ 4 & 1 & \phantom{-}3\end{bmatrix}}$
(c) $ \displaystyle{\begin{bmatrix}\phantom{-}2 & 2 & \phantom{-}3\\ -3 & 1 & \phantom{-}3\\ -1 & 5 & -6\end{bmatrix}}$
(d) $ \displaystyle{\begin{bmatrix}\phantom{-}6 & 10 & -3\\ -1 & 3 & \phantom{-}5\\ \phantom{-}5 & 3 & \phantom{-}2\end{bmatrix}}$
(e) $ \displaystyle{\begin{bmatrix}-3 & 4 & 11\\ 2 & 6 & 1\\ 11 & 16 & -6\end{bmatrix}}$
(f) $ \displaystyle{\begin{bmatrix}7 & 11 & -3\\ 5 & -5 & 12\\1 & 7 & -5\end{bmatrix}}$
5. (a) $ 5 $
(b) $ -4 $
(c) $ 175 $
(d) $ 17 $
(e) $ -143$
(f) $ 300 $
7. $ 32 $
8. $ 8 $
14. This is the set of all points which satisfy $ x - y - z = 0 $.