Let's solve each problem step-by-step.

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Problem 7: $\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$

Given:
$\mathbf{u} = 4\mathbf{i} + \mathbf{j}, \quad \mathbf{v} = \mathbf{i} - 3\mathbf{j}, \quad \mathbf{w} = 3\mathbf{i} + 2\mathbf{j}$

Step 1: Compute $\mathbf{u} \cdot \mathbf{v}$

The dot product is given by: $\mathbf{u} \cdot \mathbf{v} = (4)(1) + (1)(-3) = 4 - 3 = 1$

Step 2: Compute $\mathbf{u} \cdot \mathbf{w}$

Similarly, for $\mathbf{u} \cdot \mathbf{w}$: $\mathbf{u} \cdot \mathbf{w} = (4)(3) + (1)(2) = 12 + 2 = 14$

Step 3: Add the results

$\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} = 1 + 14 = 15$

Answer: $15$

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Problem 8: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})$

Given:
$\mathbf{u} = 5\mathbf{i} + \mathbf{j}, \quad \mathbf{v} = \mathbf{i} - 3\mathbf{j}, \quad \mathbf{w} = 3\mathbf{i} + 7\mathbf{j}$

Step 1: Compute $\mathbf{v} + \mathbf{w}$

Add the components of $\mathbf{v}$ and $\mathbf{w}$: $\mathbf{v} + \mathbf{w} = (\mathbf{i} - 3\mathbf{j}) + (3\mathbf{i} + 7\mathbf{j}) = (1 + 3)\mathbf{i} + (-3 + 7)\mathbf{j} = 4\mathbf{i} + 4\mathbf{j}$

Step 2: Compute $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})$

The dot product is: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = (5)(4) + (1)(4) = 20 + 4 = 24$

Answer: $24$

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Problem 9: $(\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v})$

Given:
$\mathbf{u} = 5\mathbf{i} + \mathbf{j}, \quad \mathbf{v} = \mathbf{i} - 3\mathbf{j}$

Step 1: Compute $\mathbf{u} + \mathbf{v}$

Add the components: $\mathbf{u} + \mathbf{v} = (5\mathbf{i} + \mathbf{j}) + (\mathbf{i} - 3\mathbf{j}) = (5 + 1)\mathbf{i} + (1 - 3)\mathbf{j} = 6\mathbf{i} - 2\mathbf{j}$

Step 2: Compute $\mathbf{u} - \mathbf{v}$

Subtract the components: $\mathbf{u} - \mathbf{v} = (5\mathbf{i} + \mathbf{j}) - (\mathbf{i} - 3\mathbf{j}) = (5 - 1)\mathbf{i} + (1 + 3)\mathbf{j} = 4\mathbf{i} + 4\mathbf{j}$

Step 3: Compute the dot product

$(\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = (6)(4) + (-2)(4) = 24 - 8 = 16$

Answer: $16$

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Summary of Answers:

1. $\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} = 15$
2. $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = 24$
3. $(\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = 16$

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Do you want details or have any questions? Here are 5 related problems you could try:

1. Compute $\mathbf{u} \cdot \mathbf{w}$ where $\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}$ and $\mathbf{w} = 4\mathbf{i} + 5\mathbf{j}$.
2. Verify if $\mathbf{v} \cdot \mathbf{w} = 0$ implies perpendicularity.
3. Find $\| \mathbf{u} \|$ where $\mathbf{u} = 7\mathbf{i} + 4\mathbf{j}$.
4. Evaluate $(\mathbf{u} - \mathbf{w}) \cdot (\mathbf{u} + \mathbf{w})$.
5. Determine the angle between $\mathbf{v}$ and $\mathbf{w}$.

Tip: Remember, the dot product is zero when two vectors are perpendicular!