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

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

Step 1: Understand that A = (x, y, z) and B = (1, 2, 3).
Step 2: The dot product A · B is calculated as x*1 + y*2 + z*3.
Step 3: Set up the equation from the dot product: x*1 + y*2 + z*3 = 14.
Step 4: Rewrite the equation: x + 2y + 3z = 14.
Step 5: Let k = x + y + z, which means we want to find the value of k.
Step 6: We can express z in terms of k: z = k - x - y.
Step 7: Substitute z back into the equation: x + 2y + 3(k - x - y) = 14.
Step 8: Simplify the equation: x + 2y + 3k - 3x - 3y = 14.
Step 9: Combine like terms: -2x - y + 3k = 14.
Step 10: Rearrange to find k: 3k = 14 + 2x + y.
Step 11: Since x and y can take various values, we can find multiple solutions for k.

Dot Product – Understanding the dot product of two vectors and how to set up the equation based on given values.
Linear Equations – Solving for variables in a linear equation derived from the dot product.
Variable Relationships – Recognizing the relationship between the sum of variables and their individual contributions to the dot product.