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
Step 1: Calculate the required values for Part 1. Given vectors U(2,1,3) and V(4,-5,1). 1) Calculate U · V The dot product of two vectors U = (u_x, u_y, u_z) and V = (v_x, v_y, v_z) is u_x v_x + u_y v_y + u_z v_z. U · V = (2)(4) + (1)(-5) + (3)(1) = 8 - 5 + 3 = 6 2) Calculate ||U||^2 The squared magnitude of a vector U is ||U||^2 = u_x^2 + u_y^2 + u_z^2. ||U||^2 = 2^2 + 1^2 + 3^2 = 4 + 1 + 9 = 14 3) Calculate ||V|| The magnitude of a vector V is ||V|| = sqrt(v_x^2 + v_y^2 + v_z^2). ||V|| = sqrt(4^2 + (-5)^2 + 1^2) = sqrt(16 + 25 + 1) = sqrt(42) = sqrt(42) 4) Calculate (U, V) The cosine of the angle between two vectors is given by () = U · V||U|| · ||V||. First, find ||U|| = sqrt(||U)||^2 = sqrt(14). (U, V) = (6)/(sqrt(14) · 42) = (6)/(sqrt(14 · 42)) = (6)/(sqrt(14 · 3 · 14)) = (6)/(sqrt(14^2 · 3)) = (6)/(14sqrt(3)) Simplify the fraction and rationalize the denominator: (U, V) = (3)/(7sqrt(3)) = 3sqrt(3)7sqrt(3)sqrt(3) = 3sqrt(3)7 · 3 = sqrt(3)7 = sqrt(3)7 5) Calculate ||U + V|| First, calculate the vector sum U + V: U + V = (2+4, 1+(-5), 3+1) = (6, -4, 4) Now, calculate its magnitude: ||U + V|| = sqrt(6^2 + (-4)^2 + 4^2) = sqrt(36 + 16 + 16) = sqrt(68) Simplify the radical: sqrt(68) = sqrt(4 · 17) = 2sqrt(17) = 2sqrt(17) Step 2: Calculate the required values for Part 2. Given points A(-4,2,-1), B(-1,-2,4), and C(-2,3,1). 1) Calculate AB The vector AB is found by subtracting the coordinates of A from B. AB = B - A = (-1 - (-4), -2 - 2, 4 - (-1)) = (3, -4, 5) = (3, -4, 5) 2) Calculate AC The vector AC is found by subtracting the coordinates of A from C. AC = C - A = (-2 - (-4), 3 - 2, 1 - (-1)) = (2, 1, 2) = (2, 1, 2) 3) Calculate AB · AC Using the vectors calculated above: AB · AC = (3)(2) + (-4)(1) + (5)(2) = 6 - 4 + 10 = 12