If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value

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Question: If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.

Options:

Correct Answer: -1

Solution:

A · B = 3k - 8 - 2 = 0; 3k - 10 = 0; k = 10/3.

If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.

If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.

Step 1: Understand that two vectors are orthogonal if their dot product is zero.
Step 2: Write down the formula for the dot product of vectors A and B. For A = (3, -2, 1) and B = (k, 4, -2), the dot product A · B is calculated as: A · B = (3 * k) + (-2 * 4) + (1 * -2).
Step 3: Substitute the values into the dot product formula: A · B = 3k - 8 - 2.
Step 4: Simplify the expression: A · B = 3k - 10.
Step 5: Set the dot product equal to zero because the vectors are orthogonal: 3k - 10 = 0.
Step 6: Solve for k by adding 10 to both sides: 3k = 10.
Step 7: Divide both sides by 3 to find k: k = 10/3.

Orthogonal Vectors – Vectors are orthogonal if their dot product is zero.
Dot Product Calculation – The dot product of two vectors is calculated by multiplying corresponding components and summing the results.
Solving Linear Equations – Finding the value of a variable in a linear equation.