First slide

Algebra of vectors

Question

Let ABC be a triangle, the position vectors of whose vertices are respectively $\hat{i} + 2 \hat{j} + 4 \hat{k}$ $- 2 \hat{i} + 2 \hat{j} + \hat{k} and 2 \hat{i} + 4 \hat{j} - 3 \hat{k}$ then $∆ A B C$ is

Moderate

A

isosceles
B

equilateral
C

right angled
D

none of these

Solution

$\begin{array}{l} \vec{BC} = \vec{OC} - \vec{OB} = 4 \hat{i} + 2 \hat{j} - 4 \hat{k} \\ \vec{AB} = - 3 \hat{i} - 3 \hat{k}, \vec{AC} = \hat{i} + 2 \hat{j} - 7 \hat{k} \end{array}$
$\begin{array}{l} {BC}^{2} = 36, {AB}^{2} = 18, {AC}^{2} = 54 \\ Clearly, {AC}^{2} = {BC}^{2} + {AB}^{2} \\ ∴ ∠ B = 90^{\circ} \end{array}$

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