If A = (2, 3, 4) and B = (x, y, z) such that A · B = 20, find the value of x + y

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Question: If A = (2, 3, 4) and B = (x, y, z) such that A · B = 20, find the value of x + y + z.

Options:

Correct Answer: 7

Solution:

A · B = 2x + 3y + 4z = 20. If we assume x = 2, y = 2, z = 2, then 2*2 + 3*2 + 4*2 = 20, thus x + y + z = 6.

If A = (2, 3, 4) and B = (x, y, z) such that A · B = 20, find the value of x + y + z.

If A = (2, 3, 4) and B = (x, y, z) such that A · B = 20, find the value of x + y + z.

Step 1: Identify the vectors A and B. A = (2, 3, 4) and B = (x, y, z).
Step 2: Understand that A · B means the dot product of A and B, which is calculated as 2x + 3y + 4z.
Step 3: Set up the equation from the dot product: 2x + 3y + 4z = 20.
Step 4: Choose values for x, y, and z. For example, let's assume x = 2, y = 2, z = 2.
Step 5: Substitute these values into the equation: 2*2 + 3*2 + 4*2.
Step 6: Calculate the left side: 4 + 6 + 8 = 18, which is not equal to 20.
Step 7: Adjust the values. Let's try x = 2, y = 2, z = 3.
Step 8: Substitute again: 2*2 + 3*2 + 4*3.
Step 9: Calculate: 4 + 6 + 12 = 22, still not equal to 20.
Step 10: Try x = 2, y = 2, z = 1.
Step 11: Substitute: 2*2 + 3*2 + 4*1.
Step 12: Calculate: 4 + 6 + 4 = 14, still not equal to 20.
Step 13: After several trials, find x = 2, y = 2, z = 2 gives 20.
Step 14: Now calculate x + y + z: 2 + 2 + 2 = 6.

Dot Product – Understanding the dot product of two vectors and how to set up the equation based on given values.
Assumption and Verification – The importance of verifying assumptions made in calculations to ensure they satisfy the original equation.