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 are the solutions to questions (3) and (4) from the image. 3) Given PQ = 2 \\ 3 and OQ = -1 \\ 4 . Find the coordinates of P. Step 1: Use the vector relationship. The displacement vector PQ is defined as the position vector of Q minus the position vector of P: PQ = OQ - OP To find the coordinates of P, we need to find OP. Rearrange the equation: OP = OQ - PQ Step 2: Substitute the given vectors. OP = -1 \\ 4 - 2 \\ 3 Step 3: Perform the vector subtraction. Subtract the corresponding components: OP = -1 - 2 \\ 4 - 3 Step 4: Calculate the resulting vector. OP = -3 \\ 1 Step 5: State the coordinates of P. The position vector OP gives the coordinates of point 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 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. Multiply each component of vector b by 2: 2 m \\ -3 = 2m \\ -6 Step 4: Perform the vector addition. Substitute the result back and add the corresponding components: 2 \\ 5 + 2m \\ -6 = 8 \\ n 2 + 2m \\ 5 + (-6) = 8 \\ n 2 + 2m \\ -1 = 8 \\ n Step 5: Equate the corresponding components. For the vectors to be equal, their x-components must be equal and their y-components must be equal. 2 + 2m = 8 -1 = n Step 6: Solve for m and n. From the y-components, we get: n = -1 From the x-components: 2m = 8 - 2 2m = 6 m = (6)/(2) m = 3 The values are m = 3, n = -1. 3 done, 2 left today. You're making progress.