If the vector a = (1, 2) and b = (3, 4), find the angle between them using the dot product.

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Q: If the vector a = (1, 2) and b = (3, 4), find the angle between them using the dot product.

Solution: cos(θ) = (a · b) / (|a| |b|). a · b = 1*3 + 2*4 = 11, |a| = √(1^2 + 2^2) = √5, |b| = √(3^2 + 4^2) = 5. Thus, cos(θ) = 11 / (√5 * 5) = 11 / (5√5), θ = 60 degrees.

Steps: 12

Step 1: Identify the vectors a and b. Here, a = (1, 2) and b = (3, 4).
Step 2: Calculate the dot product of a and b. Use the formula a · b = a1*b1 + a2*b2. So, a · b = 1*3 + 2*4.
Step 3: Perform the multiplication: 1*3 = 3 and 2*4 = 8. Now add them together: 3 + 8 = 11. Therefore, a · b = 11.
Step 4: Calculate the magnitude of vector a. Use the formula |a| = √(a1^2 + a2^2). So, |a| = √(1^2 + 2^2).
Step 5: Calculate 1^2 = 1 and 2^2 = 4. Now add them: 1 + 4 = 5. Therefore, |a| = √5.
Step 6: Calculate the magnitude of vector b. Use the formula |b| = √(b1^2 + b2^2). So, |b| = √(3^2 + 4^2).
Step 7: Calculate 3^2 = 9 and 4^2 = 16. Now add them: 9 + 16 = 25. Therefore, |b| = √25 = 5.
Step 8: Use the cosine formula to find cos(θ). The formula is cos(θ) = (a · b) / (|a| * |b|). Substitute the values: cos(θ) = 11 / (√5 * 5).
Step 9: Calculate the denominator: |a| * |b| = √5 * 5. This is equal to 5√5.
Step 10: Now, substitute this back into the cosine formula: cos(θ) = 11 / (5√5).
Step 11: To find the angle θ, use the inverse cosine function: θ = cos⁻¹(11 / (5√5)).
Step 12: Calculate θ using a calculator to find that θ is approximately 60 degrees.

If the vector a = (1, 2) and b = (3, 4), find the angle between them using the dot product.

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