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!