Question

Use vectors to determine whether the points are collinear. (0,0,0),(1,3,-2),(2,-6,4)

Short answer

No, the points (0,0,0),(1,3,-2),(2,-6,4) are not collinear.

Step-by-step solution

Create vectors between the points

Firstly, create vectors based on the given points. These vectors can be created by taking the difference between the position vectors of these points. For instance, to create vector AB: AB = B - A vector AB = (1,3,-2) - (0,0,0) => AB = (1,3,-2) Similarly, to create vector AC: AC = C - A vector AC = (2,-6,4) - (0,0,0) => AC = (2,-6,4)

Finding if the vectors are scalar multiples of each other

The vectors AB = (1,3,-2) and AC = (2,-6,4) are to be compared to see if they're scalar multiples of each other. This could be done by dividing the corresponding components of each vector. So we check if ABx/ACx = ABy/ACy = ABz/ACz. Therefore we compare: 1/2 = 3/-6 = -2/4 But as it can be seen, none of these comparisons are equal to each other hence AB and AC aren't scalar multiples of each other.

Conclude if the points are collinear or not.

As the vectors AB and AC are not scalar multiples of each other and it's concluded that the original points (0,0,0),(1,3,-2),(2,-6,4) aren't collinear.