Question

Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6).

Answer 1

The given points are P=( 4,2,−6 ) and Q=( 10,−16,6 ) .

Let A=( a,b,c ) and B=( x,y,z ) be the coordinates of the points which trisect the line segment joining the pointsP and Q.

Therefore, the ratio AP:AQ=1:2 and BP:BQ=2:1 .

According to section formula,

A( a,b,c )=( m x 2 +n x 1 m+n , m y 2 +n y 1 m+n , m z 2 +n z 1 m+n )

Therefore,

a= ( 1⋅10+2⋅4 ) 1+2 = 10+8 3 = 18 3 =6

b= 1⋅( −16 )+2⋅2 1+2 = −16+4 3 = −12 3 =−4

c= 1⋅6+2⋅( −6 ) 1+2 = 6−12 3 = −6 3 =−2

Hence, A=( 6,−4,−2 ) .

According to section formula,

B( 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 )

Therefore,

x= ( 2⋅10+1⋅4 ) 2+1 = 20+4 3 = 24 3 =8

y= ( 2⋅( −16 )+1⋅2 ) 2+1 = −32+2 3 = −30 3 =−10

z= ( 2⋅6+1⋅( −6 ) ) 2+1 = 12−6 3 = 6 3 =2

Hence, B=( 8,−10,2 ) .

Therefore, the coordinates of the points which trisect the line segment joining the points P and Q are A=( 6,−4,−2 ) and B=( 8,−10,2 ) .