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 vector p. Given p = OA + (1)/(2) OB + 3OC. Substitute the given vectors: p = 5 \\ -2 + (1)/(2) 2 \\ 4 + 3 -4 \\ 2 Step 2: Perform scalar multiplication. p = 5 \\ -2 + (1)/(2) × 2 \\ (1)/(2) × 4 + 3 × -4 \\ 3 × 2 p = 5 \\ -2 + 1 \\ 2 + -12 \\ 6 Step 3: Add the vectors. p = 5 + 1 - 12 \\ -2 + 2 + 6 p = 6 - 12 \\ 0 + 6 p = -6 \\ 6 Step 4: Find the magnitude of vector p. The magnitude of a vector x \\ y is sqrt(x^2 + y^2). |p| = sqrt((-6)^2 + 6^2) |p| = sqrt(36 + 36) |p| = sqrt(72) Simplify the square root: |p| = sqrt(36 × 2) = 6sqrt(2) Step 5: Determine the unit vector in the direction of p. A unit vector is given by (p)/(|p|). Unit vector = (1)/(6sqrt(2)) -6 \\ 6 Unit vector = (-6)/(6sqrt(2)) \\ (6)/(6sqrt(2)) Unit vector = (-1)/(sqrt(2)) \\ (1)/(sqrt(2)) Rationalize the denominators: Unit vector = -1 × sqrt(2)sqrt(2) × sqrt(2) \\ 1 × sqrt(2)sqrt(2) × sqrt(2) Unit vector = -sqrt(2)2 \\ sqrt(2)2 The unit vector is -sqrt(2)2 \\ sqrt(2)2 . Send me the next one 📸