Question 1083103
The sum of two vectors must equal the other vector,
{{{PQ+QR=PR}}}
So then,
{{{PQ=OP-OQ}}}
{{{QR=OQ-OR}}}
{{{PR=OP-OR}}}
where O stands for the origin,
So,
{{{PQ= (matrix(1,5,3,",",-2,",",1))+(matrix(1,5,1,",",-3,",",5))}}}
{{{PQ= (matrix(1,5,2,",",1,",",-4))}}}
.
.
{{{QR= (matrix(1,5,1,",",-3,",",5))+(matrix(1,5,2,",",1,",",4))}}}
{{{QR= (matrix(1,5,-1,",",-4,",",9))}}}
.
.
{{{PR= (matrix(1,5,3,",",-2,",",1))+(matrix(1,5,2,",",1,",",4))}}}
{{{PR= (matrix(1,5,1,",",-3,",",5))}}}
.
.
And,
{{{PQ+QR=(matrix(1,5,2,",",1,",",-4))+(matrix(1,5,-1,",",-4,",",9))}}}
{{{PQ+QR=(matrix(1,5,1,",",-3,",",5))}}}
{{{PQ+QR=PR}}}
So the two vectors do add up to the third.
Additionally, two of the vectors must be perpendicular to each other in order for that to happen.
So check the dot products,
{{{P*Q=3(1)+(-2)(-3)+1(5)=3+6+5=14}}}
{{{Q*R=1(2)+(-3)(1)+5(-4)=2-3-20=-21}}}
{{{P*R=3(2)+(-2)(1)+1(-4)=6-2-4=0}}}
Since the dot product is zero, the two vectors are perpendicular (right angle) and together with the result above it proves that the vectors form a right triangle.