If vectors A = (x, 2, 3) and B = (1, y, 4) are perpendicular, what is the value

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Question: If vectors A = (x, 2, 3) and B = (1, y, 4) are perpendicular, what is the value of x + y?

Options:

Correct Answer: 2

Solution:

A · B = x*1 + 2*y + 3*4 = 0. Thus, x + 2y + 12 = 0. Solving gives x + y = -6.

If vectors A = (x, 2, 3) and B = (1, y, 4) are perpendicular, what is the value of x + y?

If vectors A = (x, 2, 3) and B = (1, y, 4) are perpendicular, what is the value of x + y?

Step 1: Understand that two vectors are perpendicular if their dot product is zero.
Step 2: Write down the dot product formula for vectors A and B. A · B = (x, 2, 3) · (1, y, 4).
Step 3: Calculate the dot product: A · B = x*1 + 2*y + 3*4.
Step 4: Simplify the dot product: A · B = x + 2y + 12.
Step 5: Set the dot product equal to zero because the vectors are perpendicular: x + 2y + 12 = 0.
Step 6: Rearrange the equation to find x + 2y = -12.
Step 7: To find x + y, we need another equation. We can express y in terms of x: y = (-12 - x) / 2.
Step 8: Substitute y back into x + y: x + y = x + (-12 - x) / 2.
Step 9: Simplify the equation: x + y = (2x - 12 - x) / 2 = (x - 12) / 2.
Step 10: Set the equation equal to -6 to find the values of x and y: x + y = -6.

Dot Product of Vectors – The dot product of two vectors is zero if the vectors are perpendicular.
Solving Linear Equations – Finding the values of variables that satisfy a linear equation derived from the dot product.