Question

Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6) in the ratio (i) 2:3 internally, (ii) 2:3 externally.

Answer 1

The given points are A=( −2,3,5 ) and B=( 1,−4,6 ) .

Let P( x,y,z ) be the point which divides the line segment joining the given points A and B internally in the ratio 2:3 .

The coordinates of the point P which divides the line segment joining two points

A=( x 1 , y 1 , z 1 ) and B=( x 2 , y 2 , z 2 ) internally in the ratio m:n are given by

P( x,y,z )=( m x 2 +n x 1 m+n , m y 2 +n y 1 m+n , m z 2 +n z 1 m+n ) (1)

From the given points, the value of m is 2 , n is 3 , x 1 is −2 , x 2 is 1 , y 1 is 3 , y 2 is −4 , z 1 is 5 and z 2 is 6 .

Substitute the value of x 1 , x 2 , y 1 , y 2 , z 1 and z 2 in equation (1), to find P,

x= 2⋅1+3⋅( −2 ) 2+3 = 2−6 2+3 = −4 5

y= 2⋅( −4 )+3⋅3 2+3 = −8+9 2+3 = 1 5

z= 2⋅6+3⋅5 2+3 = 12+15 5 = 27 5

Thus, the point P which divides the line segment joining two points

A=( x 1 , y 1 , z 1 ) and B=( x 2 , y 2 , z 2 ) internally in the ratio 2:3 is ( −4 5 , 1 5 , 27 5 ) .

Let Q( x,y,z ) be the point which divides the line segment joining the given points A and B externally in the ratio 2:3 .

The coordinates of the point Q which divides the line segment joining two points

A=( x 1 , y 1 , z 1 ) and B=( x 2 , y 2 , z 2 ) internally in the ratio m:n are given by

Q( x,y,z )=( m x 2 +n x 1 m+n , m y 2 +n y 1 m+n , m z 2 +n z 1 m+n ) (1)

From the given points, the value of m is 2 , n is 3 , x 1 is −2 , x 2 is 1 , y 1 is 3 , y 2 is −4 , z 1 is 5 and z 2 is 6 .

Substitute the value of x 1 , x 2 , y 1 , y 2 , z 1 and z 2 in equation (1), to find Q,

x= 2⋅1−3⋅( −2 ) 2−3 = 2+6 −1 =−8

y= 2⋅( −4 )−3⋅3 2−3 = −8−9 −1 =17

z= 2⋅6−3⋅5 2−3 = 12−15 −1 = −3 −1 =3

Thus, the point Q which divides the line segment joining two points

A=( −2,3,5 ) and B=( 1,−4,6 ) externally in the ratio 2:3 is ( −8,17,3 ) .