Askiitians Tutor Team
Last Activity: 13 Days ago
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:
(3.37 m, -12.55 m)
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:
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:
(-41.04 km/h, 112.76 km/h)
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:
13 at a bearing of 67.38°
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:
32.88 at a bearing of 199.74°
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.