Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).

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What’s inside this PDF?

Question: Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).

Options:

0°
45°
60°
90°

Correct Answer: 60°

Solution:

cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 2*0 + 2*2 = 6. |A| = √(1^2 + 2^2 + 2^2) = 3, |B| = √(2^2 + 0^2 + 2^2) = 2√2. cos(θ) = 6 / (3 * 2√2) = 1/√2, θ = 45°.

Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).

Practice Questions

Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).

0°
45°
60°
90°

Questions & Step-by-Step Solutions

Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).

Step 1: Identify the vectors A and B. A = (1, 2, 2) and B = (2, 0, 2).
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components of A and B and then adding them together: A · B = (1*2) + (2*0) + (2*2).
Step 3: Perform the calculations for the dot product: A · B = 2 + 0 + 4 = 6.
Step 4: Calculate the magnitude of vector A, |A|. This is done using the formula |A| = √(1^2 + 2^2 + 2^2).
Step 5: Perform the calculations for |A|: |A| = √(1 + 4 + 4) = √9 = 3.
Step 6: Calculate the magnitude of vector B, |B|. This is done using the formula |B| = √(2^2 + 0^2 + 2^2).
Step 7: Perform the calculations for |B|: |B| = √(4 + 0 + 4) = √8 = 2√2.
Step 8: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A| |B|).
Step 9: Substitute the values into the formula: cos(θ) = 6 / (3 * 2√2).
Step 10: Simplify the expression: cos(θ) = 6 / (6√2) = 1/√2.
Step 11: Find the angle θ by taking the inverse cosine: θ = cos⁻¹(1/√2).
Step 12: Determine the angle: θ = 45°.

Dot Product – Understanding how to calculate the dot product of two vectors.
Magnitude of Vectors – Calculating the magnitude (length) of vectors using the formula √(x^2 + y^2 + z^2).
Cosine of Angle – Using the cosine formula to find the angle between two vectors.

Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).

What’s inside this PDF?

Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).

Practice Questions

Questions & Step-by-Step Solutions

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