Convert w=12(cos150degressi+sin150degressj) to component form

To convert the vector \( \mathbf{w} = 12(\cos 150^\circ \mathbf{i} + \sin 150^\circ \mathbf{j}) \) to component form, follow these steps:

1. **Calculate \( \cos 150^\circ \) and \( \sin 150^\circ \)**:
   - \( \cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2} \)
   - \( \sin 150^\circ = \sin 30^\circ = \frac{1}{2} \)

2. **Multiply by 12**:
   \[
   \mathbf{w} = 12\left(-\frac{\sqrt{3}}{2} \mathbf{i} + \frac{1}{2} \mathbf{j}\right)
   \]

3. **Simplify**:
   \[
   \mathbf{w} = -6\sqrt{3} \mathbf{i} + 6 \mathbf{j}
   \]

So, the vector in component form is:

\[
\mathbf{w} = (-6\sqrt{3}, 6)
\]

Would you like more details or have any questions?

Here are some related questions:

1. What is the geometric interpretation of the vector \( \mathbf{w} \)?
2. How would you convert this vector back to magnitude and direction form?
3. Can you express the angle 150° in radians?
4. What happens if you change the angle from 150° to 210°?
5. How would you compute the dot product of this vector with another vector?

**Tip:** Understanding trigonometric values for common angles like 30°, 45°, 60°, and their corresponding angles in other quadrants can help in quickly solving problems like this.

Learn how to convert vector w = 12(cos 150° + sin 150°) to component form using trigonometric functions. This problem covers trigonometric values for angles and vector components, suitable for high school students.

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