Find the value of k if the vectors A = (1, k, 2) and B = (2, 3, 4) are perpendic

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Question: Find the value of k if the vectors A = (1, k, 2) and B = (2, 3, 4) are perpendicular.

Options:

Correct Answer: 1

Solution:

A · B = 1*2 + k*3 + 2*4 = 0. Thus, 2 + 3k + 8 = 0, so 3k = -10, k = -10/3.

Find the value of k if the vectors A = (1, k, 2) and B = (2, 3, 4) are perpendicular.

Find the value of k if the vectors A = (1, k, 2) and B = (2, 3, 4) are perpendicular.

Step 1: Understand that two vectors are perpendicular if their dot product is equal to zero.
Step 2: Write down the vectors: A = (1, k, 2) and B = (2, 3, 4).
Step 3: Calculate the dot product of A and B using the formula A · B = A1*B1 + A2*B2 + A3*B3.
Step 4: Substitute the values into the dot product formula: A · B = 1*2 + k*3 + 2*4.
Step 5: Simplify the expression: A · B = 2 + 3k + 8.
Step 6: Set the dot product equal to zero because the vectors are perpendicular: 2 + 3k + 8 = 0.
Step 7: Combine like terms: 10 + 3k = 0.
Step 8: Solve for k by isolating it: 3k = -10.
Step 9: Divide both sides by 3 to find k: k = -10/3.

Dot Product – The dot product of two vectors is zero if the vectors are perpendicular.
Algebraic Manipulation – Solving for the variable k involves rearranging and simplifying equations.