Question

Using vector method, prove that the following points are collinear:
A (−3, −2, −5), B (1, 2, 3) and C (3, 4, 7)

Answer 1

Given the points \[A\left( - 3, - 2, - 5 \right), B\left( 1, 2, 3 \right)\] and \[C\left( 3, 4, 7 \right)\].
Then,
\[\overrightarrow{AB} =\] Position vector of B - Position vector of A
\[= \hat{i} + 2 \hat{j} + 3 \hat{k} + 3 \hat{i} + 2 \hat{j} + 5 \hat{k} \]
\[ = 4 \hat{i} + 4 \hat{j} + 8 \hat{k} \]
\[ = 2\left( 2 \hat{i} + 2 \hat{j} + 4 \hat{k} \right)\]
\[\overrightarrow{BC} =\]  Position vector of C - Position vector of B
\[= 3 \hat{i} + 4 \hat{j} + 7 \hat{k} - \hat{i} - 2 \hat{j} - 3 \hat{k} \]
\[ = 2 \hat{i} + 2 \hat{j} + 4 \hat{k}\]
\[\therefore \overrightarrow{AB} = 2 \overrightarrow{BC}\]
\[\text { So, } \overrightarrow{AB} , \overrightarrow{BC}\] are parallel vectors.
But B is a point common to them.
Hence, the given points  A, B and C are collinear.