1. Convert the following vectors to Cartesian form.
   1. 13 m on a bearing of 165°
   2. 120 km/h 20° West of North
2. Convert the following vectors to direction/magnitude form, giving the direction as a bearing.
   1. [12, 5]
   2. [–31, –11]

To convert vectors into Cartesian form and then back into direction/magnitude form, we need to understand a few concepts about vector representation. Let's tackle each part of your question step by step.

Converting Vectors to Cartesian Form

Cartesian form represents a vector in terms of its horizontal (x) and vertical (y) components. The first vector is given as 13 m on a bearing of 165°. A bearing is measured clockwise from North.

Vector 1: 13 m on a bearing of 165°

To convert this to Cartesian coordinates, we can use trigonometric functions. The angle from the positive x-axis (East) can be found by subtracting the bearing from 90°:

• Angle from East = 90° - 165° = -75° (or 285° if you prefer positive angles)

Now, we can find the x and y components:

• x = 13 m * cos(285°)
• y = 13 m * sin(285°)

Calculating these:

• x ≈ 13 * 0.2588 ≈ 3.37 m
• y ≈ 13 * (-0.9659) ≈ -12.55 m

Thus, the Cartesian form of the first vector is approximately:

Vector 2: 120 km/h, 20° West of North

For this vector, we first need to determine the angle from the North. Since it is 20° West of North, we can directly use this angle:

• Angle from North = 20°

Now, we can find the components:

• x = 120 km/h * sin(20°)
• y = 120 km/h * cos(20°)

Calculating these:

• x ≈ 120 * 0.3420 ≈ 41.04 km/h
• y ≈ 120 * 0.9397 ≈ 112.76 km/h

Since the x-component is directed West, we represent it as negative:

Converting Vectors to Direction/Magnitude Form

Now, let’s convert the given Cartesian vectors into direction/magnitude form. This involves calculating the magnitude and the angle (bearing) of each vector.

Vector 1: [12, 5]

First, we calculate the magnitude:

• Magnitude = √(12² + 5²) = √(144 + 25) = √169 = 13

Next, we find the angle:

• Angle = tan⁻¹(5/12) ≈ 22.62°

Since this vector is in the first quadrant, the bearing is:

• Bearing = 90° - 22.62° ≈ 67.38°

Thus, the direction/magnitude form is:

Vector 2: [-31, -11]

For this vector, we again start with the magnitude:

• Magnitude = √((-31)² + (-11)²) = √(961 + 121) = √1082 ≈ 32.88

Next, we find the angle:

• Angle = tan⁻¹(11/31) ≈ 19.74°

This vector is in the third quadrant, so we adjust the angle to find the bearing:

• Bearing = 180° + 19.74° ≈ 199.74°

Therefore, the direction/magnitude form is:

Summary

In summary, we converted the vectors to Cartesian form as follows:

• 13 m on a bearing of 165° → (3.37 m, -12.55 m)
• 120 km/h, 20° West of North → (-41.04 km/h, 112.76 km/h)

And we converted the Cartesian vectors to direction/magnitude form:

• [12, 5] → 13 at a bearing of 67.38°
• [-31, -11] → 32.88 at a bearing of 199.74°

Understanding these conversions is crucial in physics and engineering, as it allows for the analysis of forces, velocities, and other vector quantities in a clear and systematic way.