Vector Algebra Question 194
Question: The distance of the point $ B,(\mathbf{i}+2\mathbf{j}+3\mathbf{k}) $ from the line which is passing through $ A,(4\mathbf{i}+2\mathbf{j}+2\mathbf{k}) $ and which is parallel to the vector $ \overrightarrow{C}=2\mathbf{i}+3\mathbf{j}+6\mathbf{k} $ is
[Roorkee 1993]
Options:
A) 10
B) $ \sqrt{10} $
C) 100
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
- $ BM^{2}=AB^{2}-AM^{2} $ …(i)
$ \overrightarrow{AB}=-3\mathbf{i}+0\mathbf{j}+\mathbf{k} $
$ AB^{2}={{\overrightarrow{AB}}^{2}}=9+1=10 $
$ AM= $ Projection of $ \overrightarrow{AB} $ in direction of $ \overrightarrow{C} $
$ =2\mathbf{i}+3\mathbf{j}+6\mathbf{k} $
\ $ AM=\frac{\overrightarrow{AB},.,\overrightarrow{C}}{|\overrightarrow{C}|}=\frac{(-3\mathbf{i}+0\mathbf{j}+\mathbf{k}),.,(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})}{7}=0 $
\ $ BM^{2}=10-0=10 $
$ \Rightarrow BM=\sqrt{(10)} $ , {by (i)}.
BETA
