This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the values of and for which the lines intersect, we set the position vectors equal to each other. Given lines: Line 1: r = 3i - j + k + (i + 2j - k) Line 2: r = 4i + 4j + k + (-i + j + 2k) Step 1: Equate the position vectors. 3i - j + k + (i + 2j - k) = 4i + 4j + k + (-i + j + 2k) Expand and group terms by i, j, and k: (3 + )i + (-1 + 2)j + (1 - )k = (4 - )i + (4 + )j + (1 + 2)k Step 2: Equate the coefficients of i, j, and k to form a system of linear equations. For i: 3 + = 4 - + = 1 (Equation 1) For j: -1 + 2 = 4 + 2 - = 5 (Equation 2) For k: 1 - = 1 + 2 - 2 = 0 + 2 = 0 (Equation 3) Step 3: Solve the system of equations for and . We can use Equation 1 and Equation 2. Add Equation 1 and Equation 2: ( + ) + (2 - ) = 1 + 5 3 = 6 = 2 Substitute = 2 into Equation 1: 2 + = 1 = 1 - 2 = -1 Step 4: Verify the values of and using Equation 3. Substitute = 2 and = -1 into Equation 3: + 2 = 0 2 + 2(-1) = 0 2 - 2 = 0 0 = 0 The values satisfy all three equations, confirming that the lines intersect at these values. The values are = 2 and = -1. Comparing this with the given options: A = 2 and = 1 B = 2 and = -1 C = -2 and = -1 D = -2 and = 1 The correct option is B. The final answer is B. That's 2 down. 3 left today — send the next one.