To evaluate the dot product of the two vectors $\vec{C}$ and $\vec{D}$ shown in the image, we can use the formula for the dot product of two vectors in terms of their magnitudes and the angle between them:

$\vec{C} \cdot \vec{D} = |\vec{C}| |\vec{D}| \cos \theta$

where:

• $|\vec{C}| = 2$ (the magnitude of vector $\vec{C}$),
• $|\vec{D}| = 3$ (the magnitude of vector $\vec{D}$),
• $\theta = 140^\circ$ (the angle between vectors $\vec{C}$ and $\vec{D}$).

Step-by-Step Solution

1. Convert the angle to radians (optional for calculator use): In this case, we can use $\theta = 140^\circ$ directly in degrees.

2. Calculate $\cos(140^\circ)$: We know that $\cos(140^\circ) = -\cos(40^\circ)$, because $140^\circ$ is in the second quadrant, where cosine values are negative.

   Using a calculator, $\cos(140^\circ) \approx -0.7660$.

3. Apply the dot product formula:

   $\vec{C} \cdot \vec{D} = (2)(3)(-0.7660)$

4. Calculate the result:

   $\vec{C} \cdot \vec{D} = 2 \times 3 \times -0.7660 = -4.596$

Answer

The dot product $\vec{C} \cdot \vec{D}$ is approximately $-4.596$.

Would you like further details on the concept or any related questions? Here are some you might find interesting:

1. How does the angle between two vectors affect their dot product?
2. What is the significance of a negative dot product?
3. How would the result change if the angle between the vectors were 90°?
4. Can the dot product be used to determine if two vectors are orthogonal?
5. What does a zero dot product indicate about the vectors?

Tip: When two vectors form an angle greater than 90° but less than 180°, their dot product is always negative.