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
You're on a roll — Step 1: Define two vectors forming two sides of the triangle from a common vertex. Let the vertices be P(1, 4, 6), Q(-2, 5, -1), and R(1, -1, 1). We can form vectors PQ and PR. PQ = Q - P = (-2-1, 5-4, -1-6) = (-3, 1, -7) PR = R - P = (1-1, -1-4, 1-6) = (0, -5, -5) Step 2: Calculate the cross product of the two vectors. The area of a triangle with vertices P, Q, R is given by (1)/(2) |PQ × PR|. First, calculate the cross product PQ × PR: PQ × PR = i & j & k \\ -3 & 1 & -7 \\ 0 & -5 & -5 = i((1)(-5) - (-7)(-5)) - j((-3)(-5) - (-7)(0)) + k((-3)(-5) - (1)(0)) = i(-5 - 35) - j(15 - 0) + k(15 - 0) = -40i - 15j + 15k Step 3: Calculate the magnitude of the cross product. |PQ × PR| = sqrt((-40)^2 + (-15)^2 + (15)^2) = sqrt(1600 + 225 + 225) = sqrt(2050) We can simplify sqrt(2050): sqrt(2050) = sqrt(25 × 82) = 5sqrt(82) Step 4: Calculate the area of the triangle. The area of the triangle is half the magnitude of the cross product: Area = (1)/(2) |PQ × PR| = (1)/(2) (5sqrt(82)) Area = 5sqrt(82)2 The area of the triangle is 5sqrt(82)2. What's next?