Show that the points $\mathrm{P}(-2{,}3{,}5), \mathrm{Q}(1{,}2{,}3)$ and $\mathrm{R}(7{,}0,-1)$ are collinear.

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 $\vec{PQ}$ 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 $\vec{PQ}$. 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 $\vec{QR}$ by subtracting the coordinates of point Q from point R. If points P, Q, and R are collinear, then $\vec{PQ}$ and $\vec{QR}$ must be parallel.

Step 4: Simplify vector QR

Simplifying the components, we obtain the direction vector $\vec{QR}$. Now we need to check if $\vec{PQ}$ and $\vec{QR}$ are parallel.

Step 5: Check for collinearity

We observe that $\vec{QR}$ is a scalar multiple of $\vec{PQ}$ (specifically, $\vec{QR} = 2\vec{PQ}$). Since the two vectors are parallel and share a common point Q, the points P, Q, and R must be collinear.

More from this chapter

• Find the equation of set of points P such that PA²+PB²=2 k², where A and B are the points (3,4,5) and (-1,3,-7), respectively.
• The centroid of a triangle ABC is at the point (1,1,1). If the coordinates of A and B are (3,-5,7) and (-1,7,-6), respectively, find the coordinates of the poin
• Show that the points A (1, 2, 3), B (-1, -2, -1), C (2, 3, 2) and D(4,7,6) are the vertices of a parallelogram ABCD, but it is not a rectangle.
• Are the points A (3,6,9), B (10,20,30) and C(25,-41,5), the vertices of a right angled triangle?
• Find the equation of the set of the points P such that its distances from the points A(3,4,-5) and B(-2,1,4) are equal.

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