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

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

Options:

Correct Answer: 90 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (4*-3 + 3*4) / (5*5) = 0, θ = 90 degrees.

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

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

Step 1: Identify the components of vector A and vector B. Vector A = 4i + 3j and vector B = -3i + 4j.
Step 2: Calculate the dot product of vectors A and B. The dot product A · B = (4 * -3) + (3 * 4).
Step 3: Simplify the dot product calculation. A · B = -12 + 12 = 0.
Step 4: Calculate the magnitudes of vectors A and B. |A| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5. |B| = √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5.
Step 5: Use the formula for the cosine of the angle θ between the vectors: cos(θ) = (A · B) / (|A||B|).
Step 6: Substitute the values into the formula: cos(θ) = 0 / (5 * 5) = 0 / 25 = 0.
Step 7: Find the angle θ using the inverse cosine function. If cos(θ) = 0, then θ = 90 degrees.

Dot Product – The dot product of two vectors is used to find the cosine of the angle between them.
Magnitude of Vectors – Calculating the magnitude of vectors A and B is essential for determining the angle.
Angle Calculation – Using the inverse cosine function to find the angle from the cosine value.