To find the component form of the sum of the vectors $\mathbf{u}$ and $\mathbf{v}$, we need to break each vector down into its horizontal and vertical components based on the given magnitudes and direction angles. Here’s the process:

1. Determine the components of $\mathbf{u}$:

   • Given:
     • Magnitude $|\mathbf{u}| = 66$
     • Direction angle $\theta_u = 30^\circ$
   • Components:
     • $u_x = |\mathbf{u}| \cdot \cos(\theta_u) = 66 \cdot \cos(30^\circ)$
     • $u_y = |\mathbf{u}| \cdot \sin(\theta_u) = 66 \cdot \sin(30^\circ)$

   Calculating these:

   • $\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$
   • $\sin(30^\circ) = 0.5$

   Thus,

   • $u_x = 66 \cdot 0.866 \approx 57.156$
   • $u_y = 66 \cdot 0.5 = 33$

   So, $\mathbf{u} \approx \langle 57.156, 33 \rangle$.

2. Determine the components of $\mathbf{v}$:

   • Given:
     • Magnitude $|\mathbf{v}| = 27$
     • Direction angle $\theta_v = 110^\circ$
   • Components:
     • $v_x = |\mathbf{v}| \cdot \cos(\theta_v) = 27 \cdot \cos(110^\circ)$
     • $v_y = |\mathbf{v}| \cdot \sin(\theta_v) = 27 \cdot \sin(110^\circ)$

   Calculating these:

   • $\cos(110^\circ) \approx -0.342$
   • $\sin(110^\circ) \approx 0.94$

   Thus,

   • $v_x = 27 \cdot -0.342 \approx -9.234$
   • $v_y = 27 \cdot 0.94 \approx 25.38$

   So, $\mathbf{v} \approx \langle -9.234, 25.38 \rangle$.

3. Add the components of $\mathbf{u}$ and $\mathbf{v}$:

   • $\mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y \rangle$
   • $\mathbf{u} + \mathbf{v} \approx \langle 57.156 - 9.234, 33 + 25.38 \rangle$

   Simplifying:

   • $x$-component: $57.156 - 9.234 \approx 47.922$
   • $y$-component: $33 + 25.38 \approx 58.38$

   Therefore, the component form of the sum $\mathbf{u} + \mathbf{v}$ is approximately: $\mathbf{u} + \mathbf{v} \approx \langle 47.922, 58.38 \rangle$

Would you like a detailed breakdown of any part of this solution?

Here are some follow-up questions related to this topic:

1. How do we find the magnitude of a vector given its components?
2. How can we convert a vector in component form back to magnitude and angle?
3. How does changing the angle affect the vector's direction in the coordinate plane?
4. What is the significance of negative components in a vector?
5. How can trigonometric values be used to simplify vector calculations?

Tip: Remember that the direction angle is always measured from the positive $x$-axis, counterclockwise, for standard position vectors.