Thread: Vectors: simplify equation.

Simplify equation (all are vectors):
(XY - YZ) - (ZT + XT) + (ZY - ZT)
The given answer is 3TY, but I don't know how to get that answer. Please help. For example I'd say that XY - YZ = -YX - YZ = ZX, but that's not TY... Then I'd say ZT - XT = -ZT + XT = ZX(also not TY), then ZY - ZT = TY. So that would be ZX - ZX + TY = TY (not 3TY)

Originally Posted by Evaldas

Simplify equation (all are vectors):
(XY - YZ) - (ZT + XT) + (ZY - ZT)
The given answer is 3TY, but I don't know how to get that answer. Please help. For example I'd say that XY - YZ = -YX - YZ = ZX, but that's not TY... Then I'd say ZT - XT = -ZT + XT = ZX(also not TY), then ZY - ZT = TY. So that would be ZX - ZX + TY = TY (not 3TY)

Step 1. Get rid of the parentheses: XY – YZ – ZT – XT + ZY – ZT.

Step 2. Get rid of all the minus signs, using the fact that –AB = BA.

Step 3. Group the six vectors into three pairs and combine each pair into a single vector, using the fact that AB + BC = AC (you'll have to rearrange the order of the six vectors to get them to pair up properly).

One method is to represent every vector in this expression as the difference of two vectors with the beginning at the origin O. E.g., XY = OY - OX, ZT = OT - OZ, etc. Then one has to add like terms to get 3OY - 3OT = 3TY.

Done! Thank you very much!
XY + ZY + TZ + TX + ZY + TZ = (TZ + ZY) + (TX + XY) + (TZ + ZY) = TY + TY + TY = 3TY.