If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of y?

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Q: If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of y?

Solution: A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. If x = 0, then y = -9. If x = 1, y = -10. The only integer solution is y = 3.

Steps: 11

Step 1: Understand that two vectors are orthogonal if their dot product equals zero.
Step 2: Write down the vectors: A = (x, 2, 3) and B = (4, y, 6).
Step 3: Calculate the dot product A · B using the formula: A · B = x1*x2 + y1*y2 + z1*z2.
Step 4: Substitute the components of the vectors into the dot product formula: A · B = x*4 + 2*y + 3*6.
Step 5: Simplify the expression: A · B = 4x + 2y + 18.
Step 6: Set the dot product equal to zero because the vectors are orthogonal: 4x + 2y + 18 = 0.
Step 7: Rearrange the equation to solve for y: 2y = -4x - 18.
Step 8: Divide the entire equation by 2 to simplify: y = -2x - 9.
Step 9: Choose a value for x to find corresponding values of y. For example, if x = 0, then y = -9.
Step 10: If x = 1, then y = -10. Continue testing integer values for x.
Step 11: Find that when x = 3, y = -15. The only integer solution that fits the equation is y = 3 when x = 0.