Question

Use vectors to decide whether the triangle with vertices P(1, −2, −1), Q(2, 1, −3), and R(6, −1, −4) is right-angled. Yes, it is right-angled. No, it is not right-angled.

Answer 1

Answer:

Yes, it is right-angled

Explanation:

Two vectors are orthogonal if the scalar product between them is zero. Then, we will match each pair of vertices with a vector, wich is formed with the following formula:

Given two points A and B, the vector going from A to B is

$AB=B-A=(B_{x}-A_{x},B_{y}-A_{y},B_{z} -A_{z})$

So, we calculate each component separately.

$PQ=Q-P=(2-1,1-(-2),-3-(-1))=(1,3,-2)$

$QR=R-Q=(6-2,-1-1,-4-(-3))=(4,-2,-1)$

$RP=P-R=(1-6,-2-(-1),-1-(-4))=(-5,-1,3)$

Finally, using the scalar product formula

$A*B=A_{x}* B_{x}+ A_{y}* B_{y}+ A_{z}* B_{z}$

we see if the products is zero

$PQ*QR=1*4+3(-2)+(-2)*(-1)=0$

In this case we don't even have to calculate the other products as we've found that PQ and QR form a right angle.