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 angle between two vectors a and b, we use the dot product formula: = a · b||a|| · ||b|| Step 1: Solve part (a). The given vectors are a = 2i + j - 2k and b = 4i - 3j + 12k. First, calculate the dot product a · b: a · b = (2)(4) + (1)(-3) + (-2)(12) = 8 - 3 - 24 = -19 Next, calculate the magnitudes of the vectors: ||a|| = sqrt(2^2 + 1^2 + (-2)^2) = sqrt(4 + 1 + 4) = sqrt(9) = 3 ||b|| = sqrt(4^2 + (-3)^2 + 12^2) = sqrt(16 + 9 + 144) = sqrt(169) = 13 Now, substitute these values into the formula for : = (-19)/(3 · 13) = (-19)/(39) Finally, find the angle : = ^-1((-19)/(39)) ≈ 119.153^ Rounding to one decimal place, the angle is 119.2^. Step 2: Solve part (b). Based on your magnitude calculations, the vectors are a = 3i - 5j - 2k and b = i - 6k. First, calculate the dot product a · b: a · b = (3)(1) + (-5)(0) + (-2)(-6) = 3 + 0 + 12 = 15 (Your calculation for the dot product was 10, which led to an incorrect final answer. The correct dot product is 15.) Next, calculate the magnitudes of the vectors: ||a|| = sqrt(3^2 + (-5)^2 + (-2)^2) = sqrt(9 + 25 + 4) = sqrt(38) ||b|| = sqrt(1^2 + 0^2 + (-6)^2) = sqrt(1 + 0 + 36) = sqrt(37) Now, substitute these values into the formula for : = (15)/(sqrt(38) · 37) = (15)/(sqrt(1406)) Finally, find the angle : = ^-1((15)/(sqrt(1406))) ≈ 66.433^ Rounding to one decimal place, the angle is 66.4^. Send me the next one 📸