## anonymous one year ago 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).

• This Question is Open
1. IrishBoy123

|dw:1434488512478:dw| $$\vec {OX} = \vec {OP} + \vec {PX}$$ $$= \vec {OP} + \frac{1}{2} \vec {PQ}$$ $$= \vec {OP} + \frac{1}{2}( \vec {PO} + \vec {OQ})$$ $$= \vec {OP} + \frac{1}{2}( -\vec {OP} + \vec {OQ})$$ $$= \frac{1}{2} (\vec {OP} + \vec {OQ})$$

2. IrishBoy123

sorry that drawing has really messed up, but the algebra is hopefully easy to follow

3. phi

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) |dw:1434493583798:dw|

4. phi

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 |dw:1434493938496:dw|

5. phi

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)

6. phi