Konev Linear Algebra problems

Question

Problem 12: Let $A(2,1,-1)$ and $B(8,-2,11)$ be the endpoints of a linear segment $\overline{AB}$.

Find the coordinates of the point $C$ on $\overline{AB}$ such that $\overline{AC}:\overline{CB}$ is $2:1$.

I need help to solve this problem.

Answer 1

Let $A(x_A, y_A, z_A)$ and $B(x_B, y_B, z_B)$ be the endpoints of a linear segment $AB$.

Let $C(x_C, y_C, z_C)$ be a point such that $AC:CB = \lambda$.

Then,

$$ x_C = \frac{x_A + \lambda x_B}{1+λ},\quad y_C = \frac{y_A + \lambda y_B}{1+λ}, \quad z_C = \frac{z_A + \lambda z_B}{1+λ}. $$

In your example,

• $x_A = 2, y_A = 1, z_A = -1$.
• $x_B = 8, y_B = -2, z_B = 11$.
• $λ = 2$.

$$ x_C = \frac{2 + 2\cdot 8}{1+2} = 6,\quad y_C = \frac{1 + 2\cdot (-2)}{1+2} = -1, \quad z_C = \frac{-1 + 2\cdot 11}{1+2} = 7. $$