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What is the projection of vector A = 6i + 8j onto vector B = 2i + 2j?
What is the projection of vector A = 6i + 8j onto vector B = 2i + 2j?
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What is the projection of vector A = 6i + 8j onto vector B = 2i + 2j?
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Projection of A onto B = (A · B / |B|^2) * B = ((6*2 + 8*2) / (2^2 + 2^2)) * (2i + 2j) = (28/8)(2i + 2j) = 7i + 7j.
Questions & Step-by-step Solutions
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Q
Q: What is the projection of vector A = 6i + 8j onto vector B = 2i + 2j?
Solution:
Projection of A onto B = (A · B / |B|^2) * B = ((6*2 + 8*2) / (2^2 + 2^2)) * (2i + 2j) = (28/8)(2i + 2j) = 7i + 7j.
Steps: 10
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Step 1: Identify the vectors A and B. A = 6i + 8j and B = 2i + 2j.
Step 2: Calculate the dot product of A and B, which is A · B = (6 * 2) + (8 * 2).
Step 3: Compute the dot product: A · B = 12 + 16 = 28.
Step 4: Calculate the magnitude squared of vector B, which is |B|^2 = (2^2) + (2^2).
Step 5: Compute |B|^2: |B|^2 = 4 + 4 = 8.
Step 6: Use the projection formula: Projection of A onto B = (A · B / |B|^2) * B.
Step 7: Substitute the values into the formula: Projection of A onto B = (28 / 8) * (2i + 2j).
Step 8: Simplify the fraction: 28 / 8 = 3.5.
Step 9: Multiply 3.5 by vector B: 3.5 * (2i + 2j) = (3.5 * 2)i + (3.5 * 2)j.
Step 10: Calculate the final result: 7i + 7j.
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