Question

The points whose position vectors are $2\overrightarrow{i}+3\overline{j}+4\overline{k},3\overline{i}+4\overrightarrow{j}+2\overrightarrow{k}$ and $4\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{k}$ are the vertices of

• A. An isosceles triangles
• B. right angled triangle
• C. equilateral triangle
• D. right angled isosceles triangle

Answer 1

Answer: (C)

equilateral triangle

Let $A\equiv (2,3,4)$

$B\equiv (3,4,2)$

$C\equiv (4,2,3)$

By distance formula,

$AB=\sqrt{(3-2{)}^2+(4-3{)}^2+(2-4{)}^2}$

$=\sqrt{1^2+1^2+(-2{)}^2}$

$=\sqrt{1+1+4}$

$=\sqrt{6}$ units

$BC=\sqrt{(4-3{)}^2+(2-4{)}^2+(3-2{)}^2}$

$=\sqrt{1^2+(-2{)}^2+1^2}$

$=\sqrt{1+4+1}$

$=\sqrt{6}$ units

$AC=\sqrt{(4-2{)}^2+(2-3{)}^2+(3-4{)}^2}$

$=\sqrt{2^2+(-1{)}^2+(-1{)}^2}$

$=\sqrt{4+1+1}$

$=\sqrt{6}$ units

$\therefore AB=BC=AC$

So, A, B and C are vertices of an equilateral triangle.