To evaluate the dot product of the two vectors $\vec{A}$ and $\vec{B}$ shown in the image, we can use the formula:

$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$

where:

• $|\vec{A}| = 3$
• $|\vec{B}| = 5$
• $\theta = 40^\circ$

Substituting these values, we get:

$\vec{A} \cdot \vec{B} = 3 \times 5 \times \cos(40^\circ)$

Calculating this:

1. $3 \times 5 = 15$
2. $\cos(40^\circ) \approx 0.7660$
3. $15 \times 0.7660 = 11.49$

So, the dot product $\vec{A} \cdot \vec{B} \approx 11.49$.

Would you like further details or have any questions?

Related Questions:

1. How do you find the cross product of two vectors in three-dimensional space?
2. What are the properties of the dot product in vector operations?
3. How can the angle between two vectors be calculated from their dot product?
4. What is the geometric interpretation of the dot product result?
5. How does the dot product differ from the cross product?

Tip: The dot product is positive if the angle is less than 90°, negative if greater than 90°, and zero if exactly 90°.