Question

If the position vectors of the vertices A, B, C of a triangle ABC are $7\mathbf{j} + 10\mathbf{k},$ $- \mathbf{i} + 6\mathbf{j} + 6\mathbf{k}$ and $- 4\mathbf{i} + 9\mathbf{j} + 6\mathbf{k}$ respectively, the triangle is

• A. Equilateral
• B. Isosceles
• C. Scalene
• D. Right angled and isosceles also

Answer 1

Answer: (D)

Right angled and isosceles also

Answer 2

Given, position vectors of $A,B$ and C are

$7\mathbf{j} + 10\mathbf{k}$, $- \mathbf{i} + 6\mathbf{j} + 6\mathbf{k}$ and $- 4\mathbf{i} + 9\mathbf{j} + 6\mathbf{k}$respectively.

∴ $|\overset{\rightarrow}{AB}| = | - \mathbf{i} - \mathbf{j} - 4\mathbf{k}| = \sqrt{18}$

$|\overset{\rightarrow}{BC}| = | - 3\mathbf{i} + 3\mathbf{j}| = \sqrt{18}|\overset{\rightarrow}{AC}| = | - 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}| = \sqrt{36}$

Clearly, $AB = BC$ and $(AC)^{2} = (AB)^{2} + (BC)^{2}$

Hence, triangle is right angled isosceles.