Vector Algebra Question 162
Question: Let $ \mathbf{a}=2\mathbf{i}-\mathbf{j}+\mathbf{k},\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k} $ and $ \mathbf{c}=\mathbf{i}+\mathbf{j}-2\mathbf{k} $ be three vectors. A vector in the plane of b and c whose projection on a is of magnitude $ \sqrt{2/3} $ is
[IIT 1993; Pb. CET 2004]
Options:
A) $ 2\mathbf{i}+3\mathbf{j}-3\mathbf{k} $
B) $ 2\mathbf{i}+3\mathbf{j}+3\mathbf{k} $
C) $ -,2\mathbf{i}-\mathbf{j}+5\mathbf{k} $
D) $ 2\mathbf{i}+\mathbf{j}+5\mathbf{k} $
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Answer:
Correct Answer: A
Solution:
- Any vector $ \mathbf{r} $ in the plane of $ \mathbf{b} $ and $ \mathbf{c} $ is $ \mathbf{r}=\mathbf{b}+t\mathbf{c} $ or $ \mathbf{r}=(1+t)\mathbf{i}+(2+t)\mathbf{j}-(1+2t)\mathbf{k} $ ……(i)
Projection of $ \mathbf{r} $ on $ \mathbf{a} $ is $ \sqrt{( \frac{2}{3} )}\Rightarrow \frac{\mathbf{r},.,\mathbf{a}}{|\mathbf{a}|}=\sqrt{( \frac{2}{3} )} $
or $ \frac{2(1+t)-(2+t)-(1+2t)}{\sqrt{6}}=\pm \sqrt{( \frac{2}{3} )} $
\ $ ,-t-1=\pm 2\Rightarrow t=-3,1 $
Projection in (i),we get
$ \therefore ,\mathbf{r}=-2\mathbf{i}-\mathbf{j}+5\mathbf{k} $ or $ \mathbf{r}=2\mathbf{i}+3\mathbf{j}-3\mathbf{k} $ .
BETA
