The cosine of the acute angle between r1 and r2. The direction vector for r1 is d1 = i + 2j + 2k. The direction vector for r2 is d2 = -j + 3k. Find the cosine of the acute angle between the two lines.

Hey Dany — ready when you are. 5. (b) The position vector of the point of intersection ✅ Correct. Your steps and final answer are accurate. 5. (c) The cosine of the acute angle between r_1 and r_2 ❌ Incorrect. Your calculation for the dot product d_1 · d_2 is incorrect. The components were multiplied incorrectly. Here is the corrected solution: Step 1: Identify the direction vectors of the lines. The direction vector for r_1 is d_1 = i + 2j + 2k. The direction vector for r_2 is d_2 = -j + 3k. Step 2: Calculate the dot product of the direction vectors. d_1 · d_2 = (1)(0) + (2)(-1) + (2)(3) d_1 · d_2 = 0 - 2 + 6 d_1 · d_2 = 4 Step 3: Calculate the magnitudes of the direction vectors. |d_1| = sqrt(1^2 + 2^2 + 2^2) = sqrt(1+4+4) = sqrt(9) = 3 |d_2| = sqrt(0^2 + (-1)^2 + 3^2) = sqrt(0+1+9) = sqrt(10) Step 4: Use the dot product formula to find the cosine of the angle. = |d_1 · d_2||d_1||d_2| = (|4|)/((3)(sqrt(10))) = (4)/(3sqrt(10)) Rationalize the denominator: = 4sqrt(10)3sqrt(10)sqrt(10) = 4sqrt(10)30 = 2sqrt(10)15 The cosine of the acute angle between r_1 and r_2 is 2sqrt(10)15. Got more? Send 'em.