Elementary Linear Algebra: A Matrix Approach (Second Edition)

01-elementary-linear-algebra.md

ISBN-13: 9789353432997
ISBN-10: 9353432995
Authors: Lawrence E. Spence, Arnold J. Insel, Stephen H. Friedberg
Edition: 2
Binding: Paperback
Publisher: Pearson India
Published Year: 2019

Similar editions from other countries: 9780131871410, 9780131353978, 9780321656001, 9781292025032, 9780136001102

Errata

Chapter 1

p13. 2. (a)

$$A - B = \begin{bmatrix} 1 & -4 & 1 \\\ 1 & 1 & -6 \\\ \end{bmatrix}$$

02-answers-chapter-1.md

Page 11, Exercise 33

A swimmer is swimming northeast at 2 mph in still water.

(a) Give the velocity of the swimmer. Include a sketch.

(b) A current in a northerly direction at 1 mph affects the velocity of the swimmer. Give the new velocity and speed of the swimmer. Include a sketch.

---

(a) The swimmer is swimming northeast at 2 mph in still water. Northeast implies a 45-degree angle from both the east and north axes. The velocity components can be calculated using trigonometry:

$$v_x = 2 \cos(45^\circ) = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2}$$ $$v_y = 2 \sin(45^\circ) = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2}$$

Thus, the velocity vector of the swimmer is $\left( \sqrt{2}, \sqrt{2} \right)$ mph.

(b) When a current in the northerly direction at 1 mph is added, the new velocity vector is the sum of the swimmer's velocity and the current's velocity. The current's velocity vector is ((0, 1)) mph. Adding the components:

$$\text{New velocity} = \left( \sqrt{2} + 0, \sqrt{2} + 1 \right) = \left( \sqrt{2}, \sqrt{2} + 1 \right) \text{ mph}$$

To find the new speed, we calculate the magnitude of the resultant vector:

$$\text{Speed} = \sqrt{(\sqrt{2})^2 + (\sqrt{2} + 1)^2}$$ $$= \sqrt{2 + (\sqrt{2} + 1)^2}$$ $$= \sqrt{2 + (2 + 2\sqrt{2} + 1)}$$ $$= \sqrt{2 + 3 + 2\sqrt{2}}$$ $$= \sqrt{5 + 2\sqrt{2}}$$

Thus, the new velocity is $\left( \sqrt{2}, \sqrt{2} + 1 \right)$ mph and the speed is $\sqrt{5 + 2\sqrt{2}}$ mph.

Final Answer

(a) The velocity of the swimmer is $\boxed{\left( \sqrt{2}, \sqrt{2} \right)}$ mph.

(b) The new velocity is $\boxed{\left( \sqrt{2}, \sqrt{2} + 1 \right)}$ mph and the speed is $\boxed{\sqrt{5 + 2\sqrt{2}}}$ mph.