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 the vector question: Step 1: Use the condition for perpendicular vectors. Given that vectors a, b, and c are mutually perpendicular, their dot products are zero: a · b = 0 a · c = 0 b · c = 0 Also, the dot product of a vector with itself gives the square of its magnitude: a · a = |a|^2 b · b = |b|^2 c · c = |c|^2 Given that vectors X and Y are perpendicular, their dot product is zero: X · Y = 0 Step 2: Calculate the dot product X · Y. Substitute X = 4a - 3b + 2c and Y = 3a + 2b - 5c: X · Y = (4a - 3b + 2c) · (3a + 2b - 5c) Expand the dot product, using the mutual perpendicularity of a, b, c: X · Y = (4a · 3a) + (4a · 2b) + (4a · -5c) + (-3b · 3a) + (-3b · 2b) + (-3b · -5c) + (2c · 3a) + (2c · 2b) + (2c · -5c) X · Y = 12(a · a) + 8(a · b) - 20(a · c) - 9(b · a) - 6(b · b) + 15(b · c) + 6(c · a) + 4(c · b) - 10(c · c) Since a, b, c are mutually perpendicular, all cross-term dot products are zero: X · Y = 12|a|^2 - 6|b|^2 - 10|c|^2 Step 3: Substitute the given magnitudes and set the dot product to zero. Given |b| = sqrt(17) and |c| = 3. Since X · Y = 0: 12|a|^2 - 6(sqrt(17))^2 - 10(3)^2 = 0 12|a|^2 - 6(17) - 10(9) = 0 12|a|^2 - 102 - 90 = 0 12|a|^2 - 192 = 0 Step 4: Solve for |a|.