Question

Determine the values of \(x\) and \(y\) such that the points \((1,2,3),(4,7,1),\) and \((x, y, 2)\) are collinear (lie on a line).

Short answer

Question: Determine the values of x and y that make the points (1,2,3), (4,7,1), and (x,y,2) collinear. Answer: The values for x and y that make the points collinear are x = 5/2 and y = 9/2.

Step-by-step solution

Find the vectors

First, find the vectors between the points by subtracting the coordinates of the initial point from the coordinates of the terminal point. The vector from \((1, 2, 3)\) to \((4, 7, 1)\) is \(\langle 3, 5, -2 \rangle\). The vector from \((1, 2, 3)\) to \((x, y, 2)\) is \(\langle x-1, y-2, -1 \rangle\).

Check for scalar multiples

Check if the vectors are scalar multiples of each other by setting up the following equations: \(\frac{x-1}{3} = \frac{y-2}{5} = \frac{-1}{-2}\) Since we have the z component to be \(-1\), we can assume a scalar multiple of 2 for each component of the vector: 1. \(2(x-1) = 3\) 2. \(2(y-2) = 5\)

Solve the equations for x and y

To find the values for \(x\) and \(y\), solve the two equations we found in the previous step: 1. For x: \(2(x-1) = 3 \Rightarrow x-1 = \frac{3}{2} \Rightarrow x = \frac{3}{2} + 1 = \frac{5}{2}\) 2. For y: \(2(y-2) = 5 \Rightarrow y-2 = \frac{5}{2} \Rightarrow y = \frac{5}{2} + 2 = \frac{9}{2}\) The values for \(x\) and \(y\) that make the points \((1, 2, 3)\), \((4, 7, 1)\), and \((x, y, 2)\) collinear are \(x = \frac{5}{2}\) and \(y = \frac{9}{2}\).