Question

Two points have the coordinates P (6, -3,9) and Q (3, -6, -3). A point R divides line PQ internally in the ratio 1:2. The position vectors of P, Q and Rare p, q and r respectively.
(a) Express the position vector of p and q.
i. In terms of p and q.
ii. In terms of i, j and k
(b) Hence state the coordinates of R.

Answer 1

The position vector of P and Q is P = ( 6 i -3j +9k ) , Q = ( 3i -6 j -3k) , the coordinates of R is ( 5 , -4 , 2)

What is a vector ?

A vector is an object with both mass and magnitude.

It is given that

The coordinates of

P (6, -3,9 ) and Q (3, -6, -3)

point R divides line PQ internally in the ratio 1:2.

The position vectors of P, Q and R is given by  p, q and r respectively.

The position vector of P and Q in terms of i, j, k is is

P = ( 6 i -3j +9k ) , Q = ( 3i -6 j -3k)

The magnitude of P = $\rm \sqrt { 6 ^2 + (-3)^2 + 9^2$

|p| = 11.224

The magnitude of Q = $\rm \sqrt { 3^2 + (-6)^2 + ( -3)^2$

|q| = 7.348

The coordinate of R is

if a point (x,y,z) divides the line joining the points (x₁,y₁) ) and (x₂ ,y₂) in the ratio m:n, then

( x, y, z ) = $\rm \dfrac{ mx_2 + nx_1}{m + n} , \dfrac{ my_2 + ny_1}{m + n} , \dfrac{ mz_2 + nz_1}{m + n}$

The m : n = 1 : 2

(x,y,z) = $\rm \dfrac{ 3 + 2*6}{3} , \dfrac{ (-6) + 2*(-3)}{3} , \dfrac{ (-3) + 2*9}{3}$

(x,y,z) = 5 , -4 , 2

Therefore the coordinates of R is ( 5 , -4 , 2)

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