If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B?

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Question: If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B? (2019)

Options:

Correct Answer: 180 degrees

Exam Year: 2019

Solution:

A · B = 16 - 9 = 7. |A| = 5, |B| = 5. cos(θ) = 7/(5*5) = 0.28, θ = 180 degrees.

If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B? (2019)

If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B? (2019)

Step 1: Identify the vectors A and B. A = 4i + 3j and B = 4i - 3j.
Step 2: Calculate the dot product of A and B. A · B = (4)(4) + (3)(-3) = 16 - 9 = 7.
Step 3: Calculate the magnitude of vector A. |A| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5.
Step 4: Calculate the magnitude of vector B. |B| = √(4^2 + (-3)^2) = √(16 + 9) = √25 = 5.
Step 5: Use the dot product and magnitudes to find cos(θ). cos(θ) = (A · B) / (|A| * |B|) = 7 / (5 * 5) = 7 / 25 = 0.28.
Step 6: Find the angle θ using the inverse cosine function. θ = cos⁻¹(0.28).
Step 7: Calculate θ to find the angle between the vectors A and B.

Dot Product – The dot product of two vectors is calculated as A · B = |A| |B| cos(θ), which relates the angle between the vectors to their magnitudes and the dot product.
Magnitude of Vectors – The magnitude of a vector A = ai + bj is calculated as |A| = √(a² + b²).
Angle Between Vectors – The angle θ between two vectors can be found using the formula cos(θ) = (A · B) / (|A| |B|).