For question 1A-4 a) Let P and Q be two points in space, and X the midpoint of the line segment PQ. Let O be an arbitrary fixed point; show that as vectors, OX = 1/2(OP + OQ).
I am not sure how to follow the solution which is OX = OP + PX = OP + 1/2(PQ) = OP + 1/2(OQ−OP) = 1/2(OP + OQ).
MIT 18.02 Multivariable Calculus, Fall 2007
Stacey Warren - Expert brainly.com
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sorry that drawing has really messed up, but the algebra is hopefully easy to follow
First, in general, identical vectors have identical length and direction *and it does not matter where their tail is located*. In other words, we can translate a vector and it does not change its identity.
I assume you know that you can add vectors graphically by placing them "head to tail"
thus adding OP to PQ (to get OQ)
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we can write that as
OP + PQ = OQ
or if we subtract OP from both sides
PQ= OQ - OP
multiplying a vector by -1 flips the direction of the vector. now add head to tail
you get a vector that represents the length and direction of PQ
in the solution, hopefully by adding head to tail you see
OX = OP + PX
then by definition, PX is 1/2 the length (and same direction) as PQ, so
OX= OP + 1/2 PQ
and by the previous post PQ= OQ- OP
OX = OP + 1/2 ( OQ - OP)
distribute the 1/2
OX = OP +1/2 OQ - 1/2 OP
combine OP - 1/2 OP to get 1/2 OP
OX = 1/2 OP + 1/2 OQ
factor out 1/2
OX = 1/2( OP+OQ)
If that is not clear, see
Thank you both for the responses! It was silly not to think of it this way but glad you cleared it for me.