If A = (2, 3, 4) and B = (k, 0, -1) are perpendicular, find k.

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Question: If A = (2, 3, 4) and B = (k, 0, -1) are perpendicular, find k.

Options:

Correct Answer: -4

Solution:

A · B = 2k + 3*0 + 4*(-1) = 0. Thus, 2k - 4 = 0, k = 2.

If A = (2, 3, 4) and B = (k, 0, -1) are perpendicular, find k.

If A = (2, 3, 4) and B = (k, 0, -1) are perpendicular, find k.

Step 1: Understand that two vectors A and B are perpendicular if their dot product is equal to 0.
Step 2: Identify the components of vector A, which are (2, 3, 4).
Step 3: Identify the components of vector B, which are (k, 0, -1).
Step 4: Write the formula for the dot product of A and B: A · B = (2 * k) + (3 * 0) + (4 * -1).
Step 5: Simplify the dot product: A · B = 2k + 0 - 4.
Step 6: Set the dot product equal to 0 because the vectors are perpendicular: 2k - 4 = 0.
Step 7: Solve for k by adding 4 to both sides: 2k = 4.
Step 8: Divide both sides by 2 to find k: k = 2.

Dot Product – The dot product of two vectors is zero if the vectors are perpendicular.
Vector Components – Understanding how to break down vectors into their components is essential for calculating the dot product.