Describe how you could use the dot products of vectors to determine whether three given points are the vertice

Question

Describe how you could use the dot products of vectors to determine whether three given points are the vertices of a right triangle.

Describe how you could use the dot products of vectors to determine whether three given points are collinear

Answer 1

The dot product of two vectors, u and v, with angle θ between them is:

u • v = || u || || v || cosθ

If the vectors are perpendicular the angle between them is π/2. In that case the dot product of the vectors is zero.

u • v = || u || || v || cos(π/2) = || u || || v || * 0 = 0

If three points are collinear then the angle between vectors made from them is zero. So the dot product would be equal to the product of the magnitude of the two vectors.

u • v = || u || || v || cos(0) = || u || || v || * 1 = || u || || v ||

Answer 2

The definition of a right-angle triangle is that it has two sides at right angles. If you call the vertices on the triangle A,B & C, then we can write down the vectors v1=AB, v2=BC, v3=CA. Now if you want to see if there is a right angle in the triangle you need to calculate the dot products v1.v2, v2.v3 and v3.v1. If any of the vectors are at right angles to each other, then one of these dot products will be equal to zero, otherwise it is not a right-angled triangle.

If any of the vectors are collinear, then at least one of the dot products will be equal to plus or minus one (corresponding to parrallel or antiparallel pairs of vectors).

Answer 3

its a long time since I did anything like this so check your book for this

a.b = |a| |b| cos angle between
for right triangle one of the dot products must be zero

if they are collinear then the dot products must in each case = product of amplitudes