If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?
Practice Questions
1 question
Q1
If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?
0 degrees
30 degrees
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90 degrees
Cosine of angle θ = (A . B) / (|A| |B|) = (1*4 + 2*5 + 3*6) / (√14 * √77) = 0, hence θ = 90 degrees.
Questions & Step-by-step Solutions
1 item
Q
Q: If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?
Solution: Cosine of angle θ = (A . B) / (|A| |B|) = (1*4 + 2*5 + 3*6) / (√14 * √77) = 0, hence θ = 90 degrees.
Steps: 8
Step 1: Identify the components of vector A, which are (1, 2, 3).
Step 2: Identify the components of vector B, which are (4, 5, 6).
Step 3: Calculate the dot product of vectors A and B. This is done by multiplying corresponding components and adding them together: (1*4) + (2*5) + (3*6).
Step 4: Perform the calculations: 1*4 = 4, 2*5 = 10, and 3*6 = 18. Now add these results: 4 + 10 + 18 = 32.
Step 5: Calculate the magnitude (length) of vector A. The formula is |A| = √(1^2 + 2^2 + 3^2). Calculate: 1^2 = 1, 2^2 = 4, 3^2 = 9. So, |A| = √(1 + 4 + 9) = √14.
Step 6: Calculate the magnitude (length) of vector B. The formula is |B| = √(4^2 + 5^2 + 6^2). Calculate: 4^2 = 16, 5^2 = 25, 6^2 = 36. So, |B| = √(16 + 25 + 36) = √77.
Step 7: Use the cosine formula to find the cosine of the angle θ: cos(θ) = (A . B) / (|A| * |B|). Substitute the values: cos(θ) = 32 / (√14 * √77).
Step 8: Calculate the value of cos(θ). If the result is 0, then θ = 90 degrees.
