Question

Show that the points whose position vectors are as given below are collinear: \[3 \hat{i} - 2 \hat{j} + 4 \hat{k}, \hat{i} + \hat{j} + \hat{k}\text{ and }- \hat{i} + 4 \hat{j} - 2 \hat{k}\]

Answer 1

Let the points be A, B and C  with position vectors \[3 \hat{i} - 2 \hat{j} + 4 \hat{k}, \hat{i} + \hat{j} + \hat{k}\text{ and }- \hat{i} + 4 \hat{j} - 2 \hat{k}\] respectively. Then, 
\[\overrightarrow{AB} =\] Position vector of B - Position vector of A
\[= \hat{i} + \hat{j} + \hat{k} - 3 \hat{i} + 2 \hat{j} - 4 \hat{k} \]
\[ = - 2 \hat{i} + 3 \hat{j} - 3 \hat{k} \]
\[\overrightarrow{BC} =\] Position vector of C - Position vector of B
\[= - \hat{i} + 4 \hat{j} - 2 \hat{k} - \hat{i} - \hat{j} - \hat{k} \]
\[ = - 2 \hat{i} + 3 \hat{j} - 3 \hat{k}\]
\[\therefore \overrightarrow{AB} = \overrightarrow{BC}\]
\[So, \overrightarrow{AB}\]  and \[\overrightarrow{BC}\] are parallel vectors.But B  is a point common to them.
Hence, A, B and C are collinear.