Find the projection of vector A = (3, 4) onto vector B = (1, 2).

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Question: Find the projection of vector A = (3, 4) onto vector B = (1, 2).

Options:

Correct Answer: 2

Solution:

Projection of A onto B = (A · B) / |B|^2 * B. A · B = 3*1 + 4*2 = 11, |B|^2 = 1^2 + 2^2 = 5. Thus, projection = (11/5) * (1, 2) = (11/5, 22/5).

Find the projection of vector A = (3, 4) onto vector B = (1, 2).

Find the projection of vector A = (3, 4) onto vector B = (1, 2).

Step 1: Identify the vectors A and B. Here, A = (3, 4) and B = (1, 2).
Step 2: Calculate the dot product of A and B, denoted as A · B. This is done by multiplying the corresponding components of A and B and then adding them together: A · B = 3*1 + 4*2.
Step 3: Perform the multiplication: 3*1 = 3 and 4*2 = 8. Now add these results: 3 + 8 = 11. So, A · B = 11.
Step 4: Calculate the magnitude squared of vector B, denoted as |B|^2. This is done by squaring each component of B and then adding them together: |B|^2 = 1^2 + 2^2.
Step 5: Perform the squaring: 1^2 = 1 and 2^2 = 4. Now add these results: 1 + 4 = 5. So, |B|^2 = 5.
Step 6: Use the formula for the projection of A onto B: Projection of A onto B = (A · B) / |B|^2 * B.
Step 7: Substitute the values we found: Projection = (11 / 5) * B = (11 / 5) * (1, 2).
Step 8: Multiply the scalar (11/5) with each component of vector B: (11/5 * 1, 11/5 * 2) = (11/5, 22/5).
Step 9: The final result is the projection of vector A onto vector B, which is (11/5, 22/5).

Vector Projection – The process of finding the projection of one vector onto another, which involves calculating the dot product and the magnitude of the vector onto which the projection is being made.
Dot Product – A mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number, which is used in the projection formula.
Magnitude of a Vector – The length of a vector, calculated as the square root of the sum of the squares of its components, used to normalize the vector in the projection formula.