Question:

Find the coordinates of the points which trisect the line segment $AB$ where $A(2,1, -3)$ and $B(5, -8,3)$.

Updated On: Jul 6, 2022

$(4,-5,1)$, $(3,-2,-1)$
$(-4,5,1)$, $(3, -2,-1)$
$(-5,4,1)$, $(3,2,1)$
$(4, 5, -1)$, $(3,2,1)$

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The Correct Option is A

Solution and Explanation

Let $C$ and $D$ be two points which trisect the join of $AB$.

$\because C$ divides the join of $AB$ in the ratio $1 : 2$. $\therefore$ Coordinates of $C$ is $\left(\frac{1\times5+2\times 2}{1+2}, \frac{1\times \left(-8\right)+2\times 1}{1+2}, \frac{1\times 3+2\times \left(-3\right)}{1+2}\right)$ $= \left(3, -2,-1\right)$. Also, $D$ divides the join of $AB$ in the ratio $2 : 1$ $\therefore$ Coordinates of $D$ is $\left(\frac{2\times 5+1\times 2}{1+2}, \frac{2\times \left(-8\right)+1\times 1}{1+2}, \frac{2\times 3+1\times \left(-3\right)}{1+2}\right)$ $=\left(4, -5, 1\right)$

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Notes on introduction to three dimensional geometry

Concepts Used:

Introduction to Three Dimensional Geometry

In mathematics, Geometry is one of the most important topics. The concepts of Geometry are defined with respect to the planes. So, Geometry is divided into three categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.

Direction Cosines and Direction Ratios of Line

Let's consider line ‘L’ is passing through the three-dimensional plane. Now, x,y, and z are the axes of the plane, and α,β, and γ are the three angles the line making with these axes. These are called the plane's direction angles. So, correspondingly, we can very well say that cosα, cosβ, and cosγ are the direction cosines of the given line L.