LinAlg4.2.1

From Exampleproblems

Show that the points whose position vectors are $2\bar{i}-\bar{j}+\bar{k},\bar{i}-3\bar{j}-5\bar{k},3\bar{i}-4\bar{j}-4\bar{k}\,$ are the vertices of a right angled triangle.

Let $A=2\bar{i}-\bar{j}+\bar{k},B=\bar{i}-3\bar{j}-5\bar{k},C=3\bar{i}-4\bar{j}-4\bar{k}\,$ be the given points.

Now $\bar{AB}=-\bar{i}-2\bar{j}-6\bar{k},\bar{BC}=2\bar{i}-\bar{j}+\bar{k},\bar{AC}=\bar{i}-3\bar{j}-5\bar{k}\,$

Now $\bar{AB}+\bar{BC}=(-\bar{i}-2\bar{j}-6\bar{k})+(2\bar{i}-\bar{j}+\bar{k})=\bar{i}-3\bar{j}-5\bar{k}=\bar{AC}=-\bar{CA}\,$

$\bar{AB}+\bar{BC}+\bar{CA}=\bar{O}\,$

This shows that the the given points are the vertices of a triangle.

$\bar{BC}\cdot\bar{CA}=(2\bar{i}-\bar{j}+\bar{k})\cdot(-\bar{i}+3\bar{j}+5\bar{k})=-2-3+5=0\,$

Hence the two sides are perpendicular to each other.Therefore,ABC is a right angled triangle.

Main Page:Linear Algebra

Retrieved from "http://www.exampleproblems.com/wiki/index.php/LinAlg4.2.1"

Views

Personal tools

Create an account or log in

Search

Toolbox

Argan Oil
Natural Skin Care
Organic Skin Care