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

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

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

Dot Product – The dot product of two vectors is zero if the vectors are orthogonal.
Orthogonality – Two vectors are orthogonal if their dot product equals zero.
Algebraic Manipulation – Solving for a variable in an equation derived from the dot product.