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 relationship between displacement and position vectors. The displacement vector PQ is found by subtracting the position vector of the starting point P (OP) from the position vector of the ending point Q (OQ). PQ = OQ - OP To find the coordinates of P, we need to solve for OP. Rearrange the equation: OP = OQ - PQ Step 2: Substitute the given vectors into the rearranged equation. 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 components. OP = -3 \\ 1 Step 5: State the coordinates of P. The position vector OP = -3 \\ 1 means that the coordinates of point 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. Got more? Send 'em!