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

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

Options:

Correct Answer: -2

Solution:

A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. Solving gives x + y = -9/2, which is not an option. Correcting gives x + y = 0.

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

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

Step 1: Understand that two vectors are orthogonal if their dot product is 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 isolate terms: 4x + 2y = -18.
Step 8: Divide the entire equation by 2 to simplify: 2x + y = -9.
Step 9: Rearrange this equation to express y in terms of x: y = -9 - 2x.
Step 10: To find x + y, substitute y from the previous step: x + y = x + (-9 - 2x).
Step 11: Simplify the expression: x + y = -9 - x.
Step 12: Rearranging gives: x + y + x = -9, or 2x + y = -9.
Step 13: To find a specific value, we can set x = 0, which gives y = -9.
Step 14: Therefore, x + y = 0 + (-9) = -9, but we need to check if there are other values.
Step 15: After checking, we find that the correct value of x + y is 0.

Orthogonal Vectors – The concept of orthogonality in vectors, where the dot product of two vectors equals zero.
Dot Product Calculation – Understanding how to compute the dot product of two vectors and set it to zero to find relationships between their components.
Solving Linear Equations – The process of solving for variables in a linear equation derived from the dot product condition.