9Gcollinear


 * Vectors and Collinear Points **


 * Points are collinear ** means that the points lie on a straight line.[[image:9Gcollinear.gif width="377" height="269" align="right"]]

If A, B and C are collinear, then math \overrightarrow{AC} = n \overrightarrow{AB} math

Example The points P, Q and R are collinear with OP = 3__**i**__ + __**j**__ – __**k**__ OQ = __**i**__ – 2__**j**__ + __**k**__ OR = 2__**i**__ + p__**j**__ + q__**k**__ find p and q.

Using vector addition OR = OP + PR Since P,Q and R are collinear, PR = nPQ OR = OP + nPQ OR = OP + n(OQ – OP) OR = OP + nOQ – nOP

2__**i**__ + p__**j**__ + q__**k**__ = (3__**i**__ + __**j**__ – __**k**__) + n(__**i**__ – 2__**j**__ + k) – n(3__**i**__ + __**j**__ – __**k**__)

Equating coefficients 2 = 3 + n – 3n 2 = 3 – 2n 2n = –1 n = –0.5
 * __i__ ==> **

p = 1 – 2n – n p = 1 – 3n p = 1 + 1.5 p = 2.5
 * __j__ ==> **

q = –1 + n + n q = –1 + 2n q = –1 – 1 q = –2
 * __k__ ==> **

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