If A = (1, 2, 3) and B = (x, y, z) such that A · B = 14, find the value of x + y

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Question: If A = (1, 2, 3) and B = (x, y, z) such that A · B = 14, find the value of x + y + z.

Options:

Correct Answer: 7

Solution:

A · B = 1*x + 2*y + 3*z = 14. If we assume x = 2, y = 4, z = 2, then x + y + z = 8.

If A = (1, 2, 3) and B = (x, y, z) such that A · B = 14, find the value of x + y + z.

If A = (1, 2, 3) and B = (x, y, z) such that A · B = 14, find the value of x + y + z.

Step 1: Understand that A = (1, 2, 3) and B = (x, y, z).
Step 2: The dot product A · B is calculated as 1*x + 2*y + 3*z.
Step 3: We know that A · B = 14, so we can write the equation: 1*x + 2*y + 3*z = 14.
Step 4: To find values for x, y, and z, we can make an assumption. Let's assume x = 2.
Step 5: Substitute x = 2 into the equation: 1*2 + 2*y + 3*z = 14.
Step 6: Simplify the equation: 2 + 2*y + 3*z = 14.
Step 7: Subtract 2 from both sides: 2*y + 3*z = 12.
Step 8: Now, we can assume another value for y. Let's assume y = 4.
Step 9: Substitute y = 4 into the equation: 2*4 + 3*z = 12.
Step 10: Simplify the equation: 8 + 3*z = 12.
Step 11: Subtract 8 from both sides: 3*z = 4.
Step 12: Divide both sides by 3 to find z: z = 4/3.
Step 13: Now we have x = 2, y = 4, and z = 4/3.
Step 14: Calculate x + y + z: 2 + 4 + 4/3.
Step 15: Convert 2 and 4 to fractions: 2 = 6/3 and 4 = 12/3.
Step 16: Add the fractions: 6/3 + 12/3 + 4/3 = 22/3.

Dot Product – Understanding the dot product of two vectors and how to set up the equation based on given values.
Variable Assignment – The importance of correctly assigning values to variables to satisfy the equation.
Linear Equations – Solving for multiple variables in a linear equation context.