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We are asked to find the dot product \( \mathbf{u} \cdot \mathbf{v} \) given two sets of information. Let's recall the formula for the dot product when the magnitudes of the vectors and the angle between them are known:

\[
\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta
\]

Where:
- \( |\mathbf{u}| \) and \( |\mathbf{v}| \) are the magnitudes of vectors \( \mathbf{u} \) and \( \mathbf{v} \),
- \( \theta \) is the angle between the vectors.

### Part (a):
Given:
- \( |\mathbf{u}| = 1 \)
- \( |\mathbf{v}| = 2 \)
- The angle \( \theta = \frac{\pi}{6} \)

The cosine of \( \frac{\pi}{6} \) is:

\[
\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}
\]

Now applying the formula for the dot product:

\[
\mathbf{u} \cdot \mathbf{v} = 1 \times 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}
\]

Thus, for part (a), \( \mathbf{u} \cdot \mathbf{v} = \sqrt{3} \).

### Part (b):
Given:
- \( |\mathbf{u}| = 2 \)
- \( |\mathbf{v}| = 3 \)
- The angle \( \theta = 135^\circ \)

The cosine of \( 135^\circ \) is:

\[
\cos 135^\circ = -\frac{\sqrt{2}}{2}
\]

Now applying the formula for the dot product:

\[
\mathbf{u} \cdot \mathbf{v} = 2 \times 3 \times \left( -\frac{\sqrt{2}}{2} \right) = -3\sqrt{2}
\]

Thus, for part (b), \( \mathbf{u} \cdot \mathbf{v} = -3\sqrt{2} \).

### Summary:
- (a) \( \mathbf{u} \cdot \mathbf{v} = \sqrt{3} \)
- (b) \( \mathbf{u} \cdot \mathbf{v} = -3\sqrt{2} \)

Would you like any further details or clarification?

Here are 5 related questions:
1. How does the angle between two vectors affect their dot product?
2. Can the dot product ever be zero even if the vectors are non-zero? If so, when?
3. What happens to the dot product when the angle between two vectors is \( 90^\circ \)?
4. How can we use the dot product to find the angle between two vectors?
5. How does changing the magnitudes of the vectors influence the dot product?

**Tip**: The dot product is closely related to projection, which allows us to understand how much one vector points in the direction of another.

Use the given information to find the dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \). Given (a) \( |\mathbf{u}| = 1 \), \( |\mathbf{v}| = 2 \), angle between \( \mathbf{u} \) and \( \mathbf{v} \) is \( \pi/6 \); (b) \( |\mathbf{u}| = 2 \), \( |\mathbf{v}| = 3 \), angle between \( \mathbf{u} \) and \( \mathbf{v} \) is 135°.

Learn how to calculate the dot product of two vectors with given magnitudes and angles between them. This solution covers the dot product formula and includes step-by-step calculations for vectors with \( |\mathbf{u}| = 1 \), \( |\mathbf{v}| = 2 \), angle \( \pi/6 \), and vectors with \( |\mathbf{u}| = 2 \), \( |\mathbf{v}| = 3 \), angle 135°.

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