Show that the points P(−2,3,5),Q(1,2,3) and R(7,0,−1) are collinear.
Get the complete, step-by-step math solution for: "Show that the points {P}(-2{,}3{,}5), {Q}(1{,}2{,}3) and {R}(7{,}0,-1) are collinear.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Calculate the direction vector PQ
To show that three points are collinear, we can demonstrate that the direction vector between the first two points is parallel to the direction vector between the second and third points. First, we calculate the direction vector PQ by subtracting the coordinates of point P from point Q.
Step 2: Simplify vector PQ
Performing the subtraction, we find the components of the vector PQ. This vector represents the direction and magnitude from point P to point Q.
Step 3: Calculate the direction vector QR
Next, we calculate the direction vector QR by subtracting the coordinates of point Q from point R. If points P, Q, and R are collinear, then PQ and QR must be parallel.
Step 4: Simplify vector QR
Simplifying the components, we obtain the direction vector QR. Now we need to check if PQ and QR are parallel.
Step 5: Check for collinearity
We observe that QR is a scalar multiple of PQ (specifically, QR=2PQ). Since the two vectors are parallel and share a common point Q, the points P, Q, and R must be collinear.