Question

Question 5 Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6).

Class X1 - Maths -Introduction to Three Dimensional Geometry Page 277

Answer 1

P-----------S------------R ----------------Q
Let S and R in such a way that
PS = SR = RQ
Hence, S divides the Line PQ in the ratio 1 : 2 and R divides the Line in the ratio 2 : 1
Now, use section formula ,
(x, y, z )≡ { (mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)}

For point S ,
m : n = 1 : 2 and (x₁ , y₁ , z₁ ) = ( 4, 2 , -6) and ( x₂ , y₂ , z₂) = (10, -16, 6)
S ≡ { ( 1 × 10 + 2 × 4 )/(1 + 2), ( 1 × -16 + 2 × 2)/(1 + 2) , ( 1 × 6 + 2 × -6)/(1 + 2)}
S ≡{ (10 + 8)/3, ( -16 + 4)/3, ( 6 -12)/3 }
S≡ ( 6, -4, -2 )

For point R ,
m : n = 2 : 1 , (x₁ , y₁ , z₁ ) = ( 4, 2 , -6) and ( x₂ , y₂ , z₂) = (10, -16, 6)
R ≡ {2 × 10 + 1 × 4)/3 , ( 2 × -16 +1 × 2 )/3, ( 2 × 6 + 1 × -6 )/3 }
R ≡ { (20 + 4)/3 , ( -32 + 2)/3, ( 12 - 6)/3, }
R≡ ( 8, -10, 2 )

Hence, required points are ( 6, -4, -2) and ( 8, -10, 2)