If A = (2, 3) and B = (k, 1) are such that A · B = 10, find k.

If A = (2, 3) and B = (k, 1) are such that A · B = 10, find k.

Step 1: Identify the coordinates of points A and B. A = (2, 3) and B = (k, 1).
Step 2: Understand that A · B means the dot product of A and B. The dot product is calculated as (A_x * B_x) + (A_y * B_y).
Step 3: Substitute the values from A and B into the dot product formula: A · B = (2 * k) + (3 * 1).
Step 4: Simplify the expression: A · B = 2k + 3.
Step 5: Set the dot product equal to 10, as given in the question: 2k + 3 = 10.
Step 6: Solve for k by first isolating the term with k. Subtract 3 from both sides: 2k = 10 - 3.
Step 7: Simplify the right side: 2k = 7.
Step 8: Divide both sides by 2 to find k: k = 7 / 2.
Step 9: Simplify the fraction: k = 3.5.

Dot Product – The dot product of two vectors A and B is calculated by multiplying their corresponding components and summing the results.
Algebraic Manipulation – Solving for a variable involves rearranging equations and isolating the variable of interest.