Finding the Components of Vectors from a Diagram

Q1. Compute the x- and y-components of the vectors \( \vec{A} \), \( \vec{B} \), \( \vec{C} \), and \( \vec{D} \) as shown in the diagram.

Background

Topic: Vector Components

This question tests your ability to break down vectors into their x- and y-components using trigonometric relationships. Understanding vector components is fundamental in physics, as it allows you to analyze motion and forces in two dimensions.

Key Terms and Formulas

• Vector: A quantity with both magnitude and direction.

• Component: The projection of a vector along the x- or y-axis.

• To find components:

$V_x = V \cos \theta$

$V_y = V \sin \theta$

• \( V \) = magnitude of the vector

• \( \theta \) = angle measured counterclockwise from the x-axis

Step-by-Step Guidance

1. Identify the magnitude and angle for each vector from the diagram:

   • \( \vec{A} \): 8.00 m, angle = 270.0°

   • \( \vec{B} \): 15.0 m, angle = 30.0°

   • \( \vec{C} \): 12.0 m, angle = 205.0°

   • \( \vec{D} \): 10.0 m, angle = 143.0°

2. For each vector, use the formulas for x- and y-components:

   • $V_x = V \cos \theta$

   • $V_y = V \sin \theta$

3. Plug in the values for \( \vec{A} \):

   • $A_x = 8.00 \cos 270.0^$

   • $A_y = 8.00 \sin 270.0^$

4. Repeat for \( \vec{B} \), \( \vec{C} \), and \( \vec{D} \) using their respective magnitudes and angles.

5. Check the signs of your components based on the quadrant in which each vector lies. For example, a vector pointing downward or to the left will have negative components.

Try solving on your own before revealing the answer!

Final Answer:

\( \vec{A} \): $A_x = 0$, $A_y = -8.00$ m \( \vec{B} \): $B_x = 7.50$, $B_y = 13.0$ m \( \vec{C} \): $C_x = -10.9$, $C_y = -5.07$ m \( \vec{D} \): $D_x = -7.99$, $D_y = 6.02$ m

Each component is calculated using the magnitude and angle provided, applying the cosine for x-components and sine for y-components. The signs reflect the direction of each vector in the coordinate system.