https://r2.erweima.ai/mathbot/image/1726437993456eworqcmb.jpg

The problem in the image asks to find and draw the rectangular components of each vector provided, which involves breaking down each vector into its horizontal (x-component) and vertical (y-component) components.

To solve this, we use the following formulas based on the vector's magnitude \( |\vec{v}| \) and its angle \( \theta \) with the positive x-axis:

- \( v_x = |\vec{v}| \cdot \cos(\theta) \)  (x-component)
- \( v_y = |\vec{v}| \cdot \sin(\theta) \)  (y-component)

Let's solve each vector from the image:

### a) Vector \( \vec{v} = 6u \), \( \theta = 45^\circ \)
- Magnitude \( |\vec{v}| = 6u \)
- Angle \( \theta = 45^\circ \)

Using the formulas:
- \( v_x = 6 \cdot \cos(45^\circ) = 6 \cdot \frac{\sqrt{2}}{2} = 4.24u \)
- \( v_y = 6 \cdot \sin(45^\circ) = 6 \cdot \frac{\sqrt{2}}{2} = 4.24u \)

So, the rectangular components are \( (v_x, v_y) = (4.24u, 4.24u) \).

### b) Vector \( \vec{v} = 4u \), \( \theta = 30^\circ \)
- Magnitude \( |\vec{v}| = 4u \)
- Angle \( \theta = 30^\circ \)

Using the formulas:
- \( v_x = 4 \cdot \cos(30^\circ) = 4 \cdot \frac{\sqrt{3}}{2} = 3.46u \)
- \( v_y = 4 \cdot \sin(30^\circ) = 4 \cdot \frac{1}{2} = 2u \)

So, the rectangular components are \( (v_x, v_y) = (3.46u, 2u) \).

### c) Vector \( \vec{v} = 5u \), \( \theta = 130^\circ \)
- Magnitude \( |\vec{v}| = 5u \)
- Angle \( \theta = 130^\circ \)

Using the formulas:
- \( v_x = 5 \cdot \cos(130^\circ) = 5 \cdot (-0.6428) = -3.21u \)
- \( v_y = 5 \cdot \sin(130^\circ) = 5 \cdot 0.7660 = 3.83u \)

So, the rectangular components are \( (v_x, v_y) = (-3.21u, 3.83u) \).

### d) Vector \( \vec{v} = 5u \), \( \theta = 210^\circ \)
- Magnitude \( |\vec{v}| = 5u \)
- Angle \( \theta = 210^\circ \)

Using the formulas:
- \( v_x = 5 \cdot \cos(210^\circ) = 5 \cdot (-0.8660) = -4.33u \)
- \( v_y = 5 \cdot \sin(210^\circ) = 5 \cdot (-0.5) = -2.5u \)

So, the rectangular components are \( (v_x, v_y) = (-4.33u, -2.5u) \).

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Would you like further explanations or have any specific questions? Here are some related questions:

1. How do you determine the quadrant of a vector based on its angle?
2. Can you explain why we use sine and cosine to find the components of a vector?
3. What happens to the components if the angle is 90° or 270°?
4. How do we visualize the negative components of a vector in the coordinate plane?
5. How do vector components relate to vector addition?

**Tip**: When calculating the components of a vector, always pay attention to the angle's reference to the x-axis and its sign in the coordinate system.

Determina las componentes rectangulares de cada vector y dibújalas.

Learn how to determine the rectangular components of vectors with given magnitudes and angles. This problem involves trigonometric functions like sine and cosine to break down vectors into their x and y components. Suitable for students in Grades 10-12.

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