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 — here are the solutions to questions (3) and (4). 3) Given PQ = 2 \\ 3 and OQ = -1 \\ 4 . Find the coordinates of P. Step 1: Understand the relationship between the vectors. The vector PQ can be expressed as the position vector of Q minus the position vector of P: PQ = OQ - OP We want to find the coordinates of P, which are given by the position vector OP. Rearrange the formula to solve for OP: OP = OQ - PQ Step 2: Substitute the given vectors into the equation. OP = -1 \\ 4 - 2 \\ 3 Step 3: Perform the vector subtraction. Subtract the corresponding components: OP = -1 - 2 \\ 4 - 3 OP = -3 \\ 1 Step 4: State the coordinates of P. The coordinates of P are (-3, 1). 4) The vectors a = 2 \\ 5 , b = m \\ -3 and c = 8 \\ n are such that a + 2b = c. Find the values of m and n. Step 1: Write out the given vector equation. a + 2b = c Step 2: Substitute the given column vectors into the equation. 2 \\ 5 + 2 m \\ -3 = 8 \\ n Step 3: Perform the scalar multiplication on vector b. 2 m \\ -3 = 2 × m \\ 2 × -3 = 2m \\ -6 Step 4: Substitute the result back into the equation and perform vector addition. 2 \\ 5 + 2m \\ -6 = 8 \\ n 2 + 2m \\ 5 + (-6) = 8 \\ n 2 + 2m \\ -1 = 8 \\ n Step 5: Equate the corresponding components to form two separate equations. For the x-components: 2 + 2m = 8 For the y-components: -1 = n Step 6: Solve for m and n. From the y-components, we directly get: n = -1 From the x-components: 2m = 8 - 2 2m = 6 m = (6)/(2) m = 3 The values are m = 3, n = -1. What's next?