The position vectors of point A,B and C are 9i-10j , 4i+2j and Ki-2j respectively.Find the value of k if the point A,B and c are collinear. Help me solve this.thank u

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- Oct 5th 2012, 06:34 AMsharmalaPlease help me to solve this vector question.....
The position vectors of point A,B and C are 9i-10j , 4i+2j and Ki-2j respectively.Find the value of k if the point A,B and c are collinear. Help me solve this.thank u

- Oct 5th 2012, 07:42 AMHallsofIvyRe: Please help me to solve this vector question.....
What is the vector from A to B? What is the vector from A to C? Do you know what "colinear"

**means**? - Oct 5th 2012, 07:44 AMKrahlRe: Please help me to solve this vector question.....
Do you know what collinear means?

- Oct 5th 2012, 07:46 AMSorobanRe: Please help me to solve this vector question.....
Hello, sharmala1

Quote:

The position vectors of points A, B and C are 9i-10j , 4i+2j and Ki-2j respectively.

Find the value of k if the points A,B and c are collinear.

$\displaystyle \overrightarrow{BA} \:=\:\langle 5,\,\text{-}12\rangle $

$\displaystyle \overrightarrow{BC} \:=\:\langle (K\!-\!4),\,\text{-}4\rangle$

If $\displaystyle A,B,C$ are collinear, then: .$\displaystyle \overrightarrow{BA} \parallel \overrightarrow{BC}$

$\displaystyle \text{That is: }\:a\cdot\overrightarrow{BC} \:=\:\overrightarrow{BA}\,\text{ for some real number }a \ne0.$

We have: .$\displaystyle a\cdot\langle(K\!-\!4),\,\text{-}4\rangle \;=\;\langle5,\,\text{-}12\rangle$

. . . . . . . . .$\displaystyle \langle a(K\!-\!4),\,\text{-}4a\rangle \;=\;\langle 5,\,\text{-}12\rangle$

Hence: .$\displaystyle \begin{Bmatrix}a(K\!-\!4) &=& 5 & [1] \\ \text{-}4a &=& \text{-}12 & [2] \end{Bmatrix}$

From [2]: .$\displaystyle a\,=\,3$

Subtitute into [1]: .$\displaystyle 3(K-4) \:=\:5 \quad\Rightarrow\quad K-4\:=\:\tfrac{5}{3}$

Therefore: .$\displaystyle K \:=\:\frac{17}{3}$

- Oct 5th 2012, 08:13 AMKrahlRe: Please help me to solve this vector question.....
Equating the gradients can also be done.

(9,-10),(4,2),(k,-2)

$\displaystyle m=\frac{2-(-10)}{4-9}=\frac{-2-2}{k-4}$

and solve for k to get what Soroban got.