If vector A = 5i + 12j and vector B = 12i - 5j, what is the angle between A and

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Question: If vector A = 5i + 12j and vector B = 12i - 5j, what is the angle between A and B?

Options:

Correct Answer: 90 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (5*12 + 12*(-5)) / (√(169) * √(169)) = 0, θ = 90 degrees.

If vector A = 5i + 12j and vector B = 12i - 5j, what is the angle between A and B?

If vector A = 5i + 12j and vector B = 12i - 5j, what is the angle between A and B?

Step 1: Identify the components of vector A and vector B. Vector A = 5i + 12j and vector B = 12i - 5j.
Step 2: Calculate the dot product of vectors A and B. The dot product A · B = (5 * 12) + (12 * -5).
Step 3: Simplify the dot product calculation. A · B = 60 - 60 = 0.
Step 4: Calculate the magnitude of vector A. |A| = √(5^2 + 12^2) = √(25 + 144) = √169 = 13.
Step 5: Calculate the magnitude of vector B. |B| = √(12^2 + (-5)^2) = √(144 + 25) = √169 = 13.
Step 6: Use the formula for the cosine of the angle θ between two vectors: cos(θ) = (A · B) / (|A||B).
Step 7: Substitute the values into the formula. cos(θ) = 0 / (13 * 13) = 0.
Step 8: Find the angle θ using the inverse cosine function. Since cos(θ) = 0, θ = 90 degrees.

Dot Product – The dot product of two vectors is calculated as A · B = |A||B|cos(θ), which helps in finding the angle between them.
Magnitude of Vectors – The magnitude of a vector is calculated using the formula |A| = √(x^2 + y^2) for a vector A = xi + yj.
Angle Between Vectors – The angle between two vectors can be found using the inverse cosine function after calculating the dot product and magnitudes.