https://file.aiquickdraw.com/mathbot/file/1727208702616whx4nay3.jpg

The diagram shows two vectors, \( m \) and \( w \), both at different angles and lengths. Here's a breakdown of what I can observe from the image:

1. **Vector \( m \)** is at an angle of \( 30^\circ \) from the horizontal axis and has a magnitude of 5 units.
2. **Vector \( w \)** is at an angle of \( 60^\circ \) from the same horizontal axis and has a magnitude of 7 units.

Both vectors are arranged symmetrically around the vertical axis. To solve for their components:

### Decompose each vector into horizontal (x) and vertical (y) components:
#### For vector \( m \):
- \( m_x = m \cos(30^\circ) = 5 \times \cos(30^\circ) = 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \)
- \( m_y = m \sin(30^\circ) = 5 \times \sin(30^\circ) = 5 \times \frac{1}{2} = \frac{5}{2} \)

#### For vector \( w \):
- \( w_x = w \cos(60^\circ) = 7 \times \cos(60^\circ) = 7 \times \frac{1}{2} = \frac{7}{2} \)
- \( w_y = w \sin(60^\circ) = 7 \times \sin(60^\circ) = 7 \times \frac{\sqrt{3}}{2} = \frac{7\sqrt{3}}{2} \)

Now, to add the vectors, sum up their x and y components to get the resultant vector.

### Resultant vector components:
- \( R_x = m_x + w_x = \frac{5\sqrt{3}}{2} + \frac{7}{2} \)
- \( R_y = m_y + w_y = \frac{5}{2} + \frac{7\sqrt{3}}{2} \)

### Magnitude and direction of the resultant vector:
- The magnitude \( |R| \) of the resultant vector is:
  \[
  |R| = \sqrt{R_x^2 + R_y^2}
  \]
- The direction \( \theta \) of the resultant vector is:
  \[
  \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)
  \]

Let me know if you'd like me to compute the exact values for the resultant vector, or if you have further questions!

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**Five related questions:**
1. How do we calculate the angle between two vectors?
2. What is the process for finding the dot product of two vectors?
3. How can you verify if two vectors are perpendicular or parallel?
4. How do you determine the cross product between two 3D vectors?
5. What is the physical significance of a resultant vector?

**Tip:** Always decompose vectors into components when working in two or three dimensions. It simplifies vector addition and subtraction considerably.

Solve for vector m and vector w based on the given angles and magnitudes.

Learn how to decompose and calculate the resultant vector from two given vectors m and w with specific angles and magnitudes. This problem covers vector decomposition, trigonometric identities, and vector addition suitable for high school students in Grades 10-12.

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