To find the magnitude of the sum of two vectors $\mathbf{u}$ and $\mathbf{v}$, given their magnitudes and the angle between them, we can use the following formula:

$|\mathbf{u} + \mathbf{v}| = \sqrt{|\mathbf{u}|^2 + |\mathbf{v}|^2 + 2 \cdot |\mathbf{u}| \cdot |\mathbf{v}| \cdot \cos(\theta)}$

In this problem:

• $|\mathbf{u}| = 18$
• $|\mathbf{v}| = 18$
• $\theta = 110^\circ$

Step-by-Step Solution:

1. Substitute the values into the formula: $|\mathbf{u} + \mathbf{v}| = \sqrt{18^2 + 18^2 + 2 \cdot 18 \cdot 18 \cdot \cos(110^\circ)}$

2. Calculate each part individually:

   • $18^2 = 324$
   • So, $|\mathbf{u} + \mathbf{v}| = \sqrt{324 + 324 + 2 \cdot 18 \cdot 18 \cdot \cos(110^\circ)}$

3. Calculate $\cos(110^\circ)$: Using a calculator, $\cos(110^\circ) \approx -0.342$.

4. Substitute and solve: $|\mathbf{u} + \mathbf{v}| = \sqrt{324 + 324 + 2 \cdot 18 \cdot 18 \cdot (-0.342)}$ $= \sqrt{648 - 222.48}$ $= \sqrt{425.52}$

5. Find the square root: $|\mathbf{u} + \mathbf{v}| \approx 20.6$

Answer:

The magnitude of $\mathbf{u} + \mathbf{v}$ is approximately 20.6 (rounded to the nearest tenth).

Would you like a more detailed breakdown or have any questions?

Here are some related questions to explore this topic further:

1. How would the magnitude change if the angle $\theta$ was 90 degrees instead of 110 degrees?
2. What is the formula for finding the difference in magnitudes of two vectors?
3. How can we find the angle between two vectors if their magnitudes and resultant vector magnitude are given?
4. How does vector addition differ from scalar addition?
5. How does the angle between two vectors affect their resultant magnitude?

Tip: Remember, when two vectors are perpendicular ($\theta = 90^\circ$), their resultant magnitude can be simplified using the Pythagorean theorem.