A question about how vector direction affects equations

[sebastian tindall]

Hi there,

I have a very basic maths problem that has highlighted a hole in my knowledge. I was answering the following question:

The vertices of a triangle are A, B, C, with position vectors a, b, c . Show that the area of the triangle ABC is 1/2|b×c+c×a+a×b|.

and got the answer as 1/2|(a*c) + (-a*b) + (-b*c)|

I know that a*c = -c*a

but that would mean when I attempt to get the answer given in the question by switching the letters so it reads:

1/2|(-c*a) + (-a*b) + (-b*c)| i have a load of minus values in there? because I'm taking the modulus shown by the | on either side, does this mean I can just ignore the minus signs?

cheers for your help!

seb

 

[Mark44]

Hi there,

I have a very basic maths problem that has highlighted a hole in my knowledge. I was answering the following question:

The vertices of a triangle are A, B, C, with position vectors a, b, c . Show that the area of the triangle ABC is 1/2|b×c+c×a+a×b|.

and got the answer as 1/2|(a*c) + (-a*b) + (-b*c)|

I know that a*c = -c*a

Presumably you mean a X c = -c X a, and similar for the area formula you show. There are three different kinds of multiplication for vectors in R³: the cross product (denoted with X), the dot product (denoted with $\cdot$), and multiplication of a vector by a scalar, as in $c\vec{v}$. To avoid confusion in readers, use the standard notation for the cross product.

but that would mean when I attempt to get the answer given in the question by switching the letters so it reads:

1/2|(-c*a) + (-a*b) + (-b*c)| i have a load of minus values in there? because I'm taking the modulus shown by the | on either side, does this mean I can just ignore the minus signs?

I don't think so. Since all your vector products point in the opposite direction, as compared to the answer you showed above, I suspect that you did something wrong in your calculation. Please show how you got this result.