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anonymous
 one year ago
For question 1A4 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).
anonymous
 one year ago
For question 1A4 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).

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IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1dw:1434488512478:dw \(\vec {OX} = \vec {OP} + \vec {PX} \) \(= \vec {OP} + \frac{1}{2} \vec {PQ} \) \(= \vec {OP} + \frac{1}{2}( \vec {PO} + \vec {OQ}) \) \(= \vec {OP} + \frac{1}{2}( \vec {OP} + \vec {OQ}) \) \(= \frac{1}{2} (\vec {OP} + \vec {OQ}) \)

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1sorry that drawing has really messed up, but the algebra is hopefully easy to follow

phi
 one year ago
Best ResponseYou've already chosen the best response.0First, 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) dw:1434493583798:dw

phi
 one year ago
Best ResponseYou've already chosen the best response.0we 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 dw:1434493938496:dw

phi
 one year ago
Best ResponseYou've already chosen the best response.0in 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)

phi
 one year ago
Best ResponseYou've already chosen the best response.0If that is not clear, see https://www.khanacademy.org/math/linearalgebra/vectors_and_spaces/vectors/v/addingvectors

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Thank you both for the responses! It was silly not to think of it this way but glad you cleared it for me.
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