Year 12s: what will be the first way you research your future options? >>

Vectors Question

I'm having a bit of trouble with a vectors question and would really appreciate some help. "Three distinct non-zero vectors are given by OA = a, OB = b, and OC = c. If OA is perpendicular to BC and OB is perpendicular to CA, show that OC is perpendicular to AB."

Since OA is perpendicular to BC, a(c - b) = 0 ? ac - ab = 0

Since OB is perpendicular to CA, b(a - c) = 0 ? ba - bc = 0

Simultaneous equations:

ac - ab = 0
ba - bc = 0

- ab + ac = 0
ba - bc = 0

It's obvious that we get ac - bc = 0 ? c(a - b) = 0 ? so OC is perpendicular to AB.

But what I don't understand is how the -ab and ba vectors "cancel out". Doesn't it become -ab and ba = ba + ba = 2ba ? Or am I missing a trick here? :smile: