Compute the dot product and angle between the two vectors in degrees.
u = <minus−7,minus−10>, v = <5,minus−8>
Round to three decimal places when necessary.


### Step 1: Dot Product of Vectors \( \mathbf{u} \) and \( \mathbf{v} \)
The dot product formula for two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) is:

\[
\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2
\]

Substitute the components of vectors \( \mathbf{u} = \langle -7, -10 \rangle \) and \( \mathbf{v} = \langle 5, -8 \rangle \):

\[
\mathbf{u} \cdot \mathbf{v} = (-7)(5) + (-10)(-8)
\]
\[
= -35 + 80 = 45
\]

### Step 2: Magnitudes of the Vectors
The magnitude \( ||\mathbf{u}|| \) of a vector \( \mathbf{u} = \langle u_1, u_2 \rangle \) is given by:

\[
||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2}
\]

For \( \mathbf{u} = \langle -7, -10 \rangle \):

\[
||\mathbf{u}|| = \sqrt{(-7)^2 + (-10)^2} = \sqrt{49 + 100} = \sqrt{149} \approx 12.206
\]

For \( \mathbf{v} = \langle 5, -8 \rangle \):

\[
||\mathbf{v}|| = \sqrt{(5)^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.434
\]

### Step 3: Angle Between the Vectors
The cosine of the angle \( \theta \) between two vectors is given by:

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

Substituting the values:

\[
\cos \theta = \frac{45}{12.206 \times 9.434} \approx \frac{45}{115.169} \approx 0.391
\]

Now, find \( \theta \) by taking the inverse cosine:

\[
\theta = \cos^{-1}(0.391) \approx 67.000^\circ
\]

### Final Answers:
- **Dot product**: \( 45 \)
- **Angle**: \( 67.000^\circ \)

Would you like further details or have any questions?

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Here are 5 related questions for further practice:
1. What is the dot product of \( \mathbf{u} = \langle 3, -4 \rangle \) and \( \mathbf{v} = \langle 1, 2 \rangle \)?
2. How do you compute the angle between two vectors in radians?
3. Can the dot product be negative? What does that signify?
4. How does the magnitude of a vector relate to its direction?
5. What is the result when two vectors are orthogonal?

**Tip**: Always check if two vectors are perpendicular by seeing if their dot product equals zero!

Compute the dot product and angle between the two vectors in degrees. u = <−7,−10>, v = <5,−8>. Round to three decimal places when necessary.

Learn how to compute the dot product and angle between two vectors, u = <−7,−10> and v = <5,−8>, with a step-by-step solution. This problem uses vector algebra, the dot product formula, and the cosine of the angle between vectors. Suitable for students in Grades 10-12.

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