If A = (x, y, z) and B = (2, 2, 2), and A · B = 12, what is the value of x + y +

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Question: If A = (x, y, z) and B = (2, 2, 2), and A · B = 12, what is the value of x + y + z?

Options:

Correct Answer: 6

Solution:

A · B = 2x + 2y + 2z = 12, thus x + y + z = 6.

If A = (x, y, z) and B = (2, 2, 2), and A · B = 12, what is the value of x + y + z?

If A = (x, y, z) and B = (2, 2, 2), and A · B = 12, what is the value of x + y + z?

Step 1: Identify the vectors A and B. A = (x, y, z) and B = (2, 2, 2).
Step 2: Understand that the dot product A · B is calculated as x * 2 + y * 2 + z * 2.
Step 3: Write the equation for the dot product: A · B = 2x + 2y + 2z.
Step 4: Set the equation equal to 12, since A · B = 12: 2x + 2y + 2z = 12.
Step 5: Factor out the 2 from the left side: 2(x + y + z) = 12.
Step 6: Divide both sides of the equation by 2 to isolate (x + y + z): x + y + z = 12 / 2.
Step 7: Calculate the right side: x + y + z = 6.

Dot Product – The dot product of two vectors A and B is calculated by multiplying corresponding components and summing the results.
Algebraic Manipulation – The ability to rearrange and simplify equations to isolate variables or find relationships between them.