Vector Algebra Question 317
Question: The length of the perpendicular from the origin to the plane passing through the point a and containing the line $ \mathbf{r}=\mathbf{b}+\lambda \mathbf{c} $ is
Options:
A) $ \frac{[\mathbf{a},\mathbf{b},\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|} $
B) $ \frac{,[\mathbf{a},\mathbf{b},\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}|} $
C) $ \frac{[\mathbf{a},\mathbf{b},\mathbf{c}]}{|\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|} $
D) $ \frac{[\mathbf{a},\mathbf{b},\mathbf{c}]}{|\mathbf{c}\times \mathbf{a}+\mathbf{a}\times \mathbf{b}|} $
Show Answer
Answer:
Correct Answer: C
Solution:
- The given plane passes through $ \mathbf{a} $ and is parallel to the vectors $ \mathbf{b}-\mathbf{a} $ and $ \mathbf{c} $ . So it is normal to $ (\mathbf{b}-\mathbf{a})\times \mathbf{c} $ . Hence, its equation is $ (\mathbf{r}-\mathbf{a}).((\mathbf{b}-\mathbf{a})\times \mathbf{c})=0 $ or $ \mathbf{r}.(\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a})=[\mathbf{a},\mathbf{b},\mathbf{c},] $ The length of the perpendicular from the origin to this plane is $ \frac{[\mathbf{a},\mathbf{b},\mathbf{c}]}{|\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|} $ .
BETA
