Let's go through each problem one by one.

Problem 1: Find $a \cdot b$

You are given: $|a| = 2, \quad |b| = 4, \quad \text{angle} = 30^\circ$ The formula for the dot product using magnitudes and the angle is: $a \cdot b = |a| |b| \cos \theta$

Substituting the values: $a \cdot b = (2)(4) \cos 30^\circ$ $= 8 \times \frac{\sqrt{3}}{2}$ $= 4\sqrt{3} \approx 6.93$

Problem 2: Find $a \cdot b$

Given: $|a| = 80, \quad |b| = 30, \quad \text{angle} = \frac{3\pi}{4}$ Using the same formula: $a \cdot b = (80)(30) \cos \frac{3\pi}{4}$

Since: $\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$ $a \cdot b = 2400 \times -\frac{\sqrt{2}}{2}$ $= -1200\sqrt{2} \approx -1697.06$

Problem 3: Find the angle between vectors

Vectors: $a = (1, -7), \quad b = (-7, 24)$

Step 1: Compute Dot Product

$a \cdot b = (1)(-7) + (-7)(24)$ $= -7 - 168 = -175$

Step 2: Compute Magnitudes

$|a| = \sqrt{1^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50}$ $|b| = \sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$

Step 3: Compute Cosine of the Angle

$\cos \theta = \frac{a \cdot b}{|a||b|}$ $= \frac{-175}{\sqrt{50} \times 25}$

Approximating: $\sqrt{50} \approx 7.07$ $|a||b| = 7.07 \times 25 = 176.75$

$\cos \theta \approx \frac{-175}{176.75} \approx -0.99$

Step 4: Compute Angle

$\theta = \cos^{-1}(-0.99) \approx 171^\circ$

Would you like further explanations or have any questions?

Related Questions

1. How do you derive the dot product formula using magnitudes and angles?
2. Why is the cosine of $\frac{3\pi}{4}$ negative?
3. How do you compute the cross product of two vectors?
4. What is the significance of the dot product in physics?
5. How can the dot product be used to determine if two vectors are perpendicular?

Tip: The dot product of two perpendicular vectors is always zero.