Question

1. an automobile travels 9 miles due south, then 20 miles in a direction 40 degrees south of east, determine the resultant displacement.
2.find the sum or resultant of the following displacements: A, 10 ft northwest, B, 20 ft 30 degrees north of east; C, 35 ft due south.
3.which of the vector r = xi+yj+zk makes with the positive directions of the coordinate axes.
4.An airplane pilot sets a course due north and his speed is 150km/hr. There is a wind 40km/hr from the east. What is the actual velocity of the airplane?
5. A river flows at a steady speed of 3m/s. A man wishes to cross in a motorboat which travels at 5km/hr so as to reach a point directly across the river. In what direction should he steer?
6. if r1 = 2i-j+k, r2 = i+3j-2k, r3 = -2i+j-3k and r4 = 3i+2j+5k, find the scalars a, b, c such that r4=ar1+br2+cr3

Answer 1

Answer on Question #63070, Physics / Mechanics | Relativity

1. an automobile travels 9 miles due south, then 20 miles in a direction 40 degrees south of east, determine the resultant displacement.

Solution:

$$\alpha = 180 - 40 = 140{}^{\circ}$$

Cosine Law

$$d = \sqrt{9^2 + 20^2 - 29 \cdot 20 \cdot \cos 140{}^{\circ}} = 23,5 \text{ miles}$$

Answer: 23.5 miles

2. find the sum or resultant of the following displacements: A, 10 ft northwest, B, 20 ft 30 degrees north of east; C, 35 ft due south.

Solution:

1 units = 5 ft

At the terminal point of A place the initial point of B. At the terminal point of B place the initial point of C. The resultant D is formed by joining the initial point of A to the terminal point of C, i.e. $D = A + B + C$. Graphically the resultant is measured to have magnitude of 4.1 units = 20.5 ft and direction $60{}^{\circ}$ south of east.

Answer: 20.5 ft and $60{}^{\circ}$ south of east

3. Which of the vector $r = \frac{xi + yj + zk}{zk}$ makes with the positive directions of the coordinate axes.

Answer:

zk makes with the positive directions of the coordinate axes

4. An airplane pilot sets a course due north and his speed is $150\,\text{km/hr}$. There is a wind $40\,\text{km/hr}$ from the east. What is the actual velocity of the airplane?

Solution:

$$v = \sqrt{125^2 + 40^2} = 155 \text{ km/hr}$$

Answer: 155 km/hr

5. A river flows at a steady speed of $3\mathrm{m/s}$. A man wishes to cross in a motorboat, which travels at $5\mathrm{km/hr}$ to reach a point directly across the river. In what direction should he steer?

**Solution:**

$$\begin{array}{l} (1.4 \mathrm{m/s})^2 + (3.0 \mathrm{m/s})^2 = R^2 \\ 1.96 \mathrm{m}^2/\mathrm{s}^2 + 9 \mathrm{m}^2/\mathrm{s}^2 = R^2 \\ 10.96 \mathrm{m}^2/\mathrm{s}^2 = R^2 \\ \text{SQRT} (10.96 \mathrm{m}^2/\mathrm{s}^2) = R \\ 3.3 \mathrm{m/s} = R \\ \end{array}$$

The direction

tan (theta) = (opposite/adjacent)

tan (theta) = (1.4/3)

theta = invtan (1.4/3)

theta = 24.9 degree

Answer: 24.9 degree

6. If $r1 = 2i - j + k$, $r2 = i + 3j - 2k$, $r3 = -2i + j - 3k$ and $r4 = 3i + 2j + 5k$, find the scalars a, b, c such that $r4 = ar1 + br2 + cr3$

**Solution:**

We require

$$3i + 2j + 5k = a(2i - j + k) + b(i + 3j - 2k) + c(-2i + j - 3k) = (2a + b - 2c)i + (-a + 3b + c)j + (a - 2b - 3c)k.$$

Since

i, j, k are non-coplanar we have,

$$\begin{array}{l} 2a + b - 2c = 3, \\ -a + 3b + c = 2, \\ a - 2b - 3c = 5. \\ a = -2, \\ b = 1, \\ c = -3 \\ \end{array}$$

and

$$r4 = -2r1 + r2 - 3r3.$$

Answer: $a = -2$, $b = 1$, $c = -3$

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