To solve for the magnitude and direction of vector B in the equation A + B = C, we need to use vector components and trigonometry. Here's the step-by-step process:

Given:

• Magnitude of A: $|A| = 12.0 \, \text{m}$
• Direction of A: $40^\circ$ counterclockwise from the $+x$-axis.
• Magnitude of C: $|C| = 15.0 \, \text{m}$
• Direction of C: $20^\circ$ counterclockwise from the $-x$-axis (this is equivalent to $180^\circ - 20^\circ = 160^\circ$ relative to the $+x$-axis).

Step 1: Find the components of vectors A and C.

We can break down vectors A and C into their x and y components using trigonometry.

For vector A:

• $A_x = |A| \cdot \cos(40^\circ) = 12.0 \cdot \cos(40^\circ)$
• $A_y = |A| \cdot \sin(40^\circ) = 12.0 \cdot \sin(40^\circ)$

For vector C:

• $C_x = |C| \cdot \cos(160^\circ) = 15.0 \cdot \cos(160^\circ)$
• $C_y = |C| \cdot \sin(160^\circ) = 15.0 \cdot \sin(160^\circ)$

Step 2: Calculate components of vector B.

Since $\mathbf{A} + \mathbf{B} = \mathbf{C}$, we have:

• $B_x = C_x - A_x$
• $B_y = C_y - A_y$

Step 3: Calculate the magnitude and direction of vector B.

• Magnitude of B: $|B| = \sqrt{B_x^2 + B_y^2}$
• Angle of B relative to the $+x$-axis: $\theta_B = \tan^{-1} \left( \frac{B_y}{B_x} \right)$

Let's compute these values.import math

Given values

A_magnitude = 12.0 # magnitude of A in meters A_angle = 40 # angle of A in degrees relative to +x axis C_magnitude = 15.0 # magnitude of C in meters C_angle = 160 # angle of C in degrees relative to +x axis

Convert angles to radians for trigonometric calculations

A_angle_rad = math.radians(A_angle) C_angle_rad = math.radians(C_angle)

Components of vector A

A_x = A_magnitude * math.cos(A_angle_rad) A_y = A_magnitude * math.sin(A_angle_rad)

Components of vector C

C_x = C_magnitude * math.cos(C_angle_rad) C_y = C_magnitude * math.sin(C_angle_rad)

Components of vector B (B_x = C_x - A_x, B_y = C_y - A_y)

B_x = C_x - A_x B_y = C_y - A_y

Magnitude of vector B

B_magnitude = math.sqrt(B_x2 + B_y2)

Angle of vector B relative to +x axis (in degrees)

B_angle = math.degrees(math.atan2(B_y, B_x))

B_magnitude, B_angleThe magnitude and angle of vector B are as follows:

• Magnitude of B: $23.43 \, \text{m}$
• Angle of B relative to the $+x$-axis: $-173.67^\circ$ (which means it is slightly below the negative $x$-axis).

Would you like more details or clarifications?

Here are some related questions for further practice:

1. How would the magnitude of vector B change if vector C was doubled?
2. What happens to the angle of vector B if the angle of C is reduced by 30 degrees?
3. How would you approach this problem if vector A was vertical instead of angled?
4. What is the direction of vector A + B if vector C was rotated 90 degrees?
5. How do you convert between Cartesian and polar coordinates in vector problems?

Tip: Always check the quadrant in which a vector lies when calculating the angle to avoid sign errors in trigonometric functions.