Question

A and C are the points (1, 3, −2) and (4, −4, 4) respectively. Point B divides AC in the ratio 1: 2. Find the coordinates of B.

Answer 1

Given:

The two points are A(1, 3, −2) and C(4, −4, 4).

Point B divides AC in the ratio 1: 2.

To find:

The coordinates of B.

Solution:

If a point divides a lines segment in m:n, then the coordinates of that point are:

$Point=\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n},\dfrac{mz_2+nz_1}{m+n}\right)$

Point B divides AC in the ratio 1: 2. So, the coordinates of point B are:

$Point=\left(\dfrac{1(4)+2(1)}{1+2},\dfrac{1(-4)+2(3)}{1+2},\dfrac{1(4)+2(-2)}{1+2}\right)$

$Point=\left(\dfrac{4+2}{3},\dfrac{-4+6}{3},\dfrac{4-4}{3}\right)$

$Point=\left(\dfrac{6}{3},\dfrac{2}{3},\dfrac{0}{3}\right)$

$Point=\left(2,\dfrac{2}{3},0\right)$

Therefore, the coordinates of B are $Point=\left(2,\dfrac{2}{3},0\right)$.