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
Here's the solution for question 7: a) Find the value of a. Step 1: Write the vector equations in component form. For L_1: r_1 = (1 - 4)i + (1 + )j - 3k For L_2: r_2 = (9 + 2)i + (2 + a)j + (1 + 2)k Step 2: Since the lines intersect, their position vectors at the point of intersection are equal. Equate the corresponding components. 1 - 4 = 9 + 2 (1) 1 + = 2 + a (2) -3 = 1 + 2 (3) Step 3: Solve equation (3) for . -3 = 1 + 2 2 = -4 = -2 Step 4: Substitute = -2 into equation (1) to find . 1 - 4 = 9 + 2(-2) 1 - 4 = 9 - 4 1 - 4 = 5 -4 = 4 = -1 Step 5: Substitute = -1 and = -2 into equation (2) to find a. 1 + (-1) = 2 + a(-2) 0 = 2 - 2a 2a = 2 a = 1 The value of a is: 1 b) Find the position vector of the point of intersection of L_1 and L_2. Step 1: Substitute = -1 into the equation for L_1. r = i + j - 3k + (-1)(-4i + j) r = i + j - 3k + 4i - j r = (1+4)i + (1-1)j - 3k r = 5i + 0j - 3k The position vector of the point of intersection is: 5i - 3k c) Find the cosine of the angle between L_1 and L_2. Step 1: Identify the direction vectors of the lines. For L_1, the direction vector is d_1 = -4i + j. For L_2, the direction vector is d_2 = 2i + aj + 2k. Since a=1, d_2 = 2i + j + 2k. Step 2: Calculate the dot product of the direction vectors. d_1 · d_2 = (-4)(2) + (1)(1) + (0)(2) d_1 · d_2 = -8 + 1 + 0 = -7 Step 3: Calculate the magnitudes of the direction vectors. |d_1| = sqrt((-4)^2 + (1)^2 + (0)^2) = sqrt(16 + 1 + 0) = sqrt(17) |d_2| = sqrt((2)^2 + (1)^2 + (2)^2) = sqrt(4 + 1 + 4) = sqrt(9) = 3 Step 4: Use the formula for the cosine of the angle between two vectors, = d_1 · d_2|d_1| |d_2|. = (-7)/(sqrt(17) · 3) = (-7)/(3sqrt(17)) The cosine of the angle between L_1 and L_2 is: (-7)/(3sqrt(17)) 3 done, 2 left today. You're making progress.