If A = (a, b, c) and B = (1, 2, 3), find the value of a if A · B = 10.

If A = (a, b, c) and B = (1, 2, 3), find the value of a if A · B = 10.

Step 1: Identify the vectors A and B. A = (a, b, c) and B = (1, 2, 3).
Step 2: Understand that A · B means to multiply corresponding elements of A and B and then add them together.
Step 3: Write the equation for A · B: A · B = a*1 + b*2 + c*3.
Step 4: Substitute the values of b and c into the equation. We know b = 2 and c = 1.
Step 5: Replace b and c in the equation: A · B = a*1 + 2*2 + 1*3.
Step 6: Calculate the values: A · B = a + 4 + 3.
Step 7: Combine the numbers: A · B = a + 7.
Step 8: Set the equation equal to 10: a + 7 = 10.
Step 9: Solve for a by subtracting 7 from both sides: a = 10 - 7.
Step 10: Calculate the final value: a = 3.

Dot Product – The dot product of two vectors is calculated by multiplying corresponding components and summing the results.
Algebraic Manipulation – Solving for a variable in an equation by isolating it through algebraic operations.