Vector Algebra Question 128
Question: If a, b and c be three non-zero vectors, no two of which are collinear. If the vector $ \mathbf{a}+2\mathbf{b} $ is collinear with c and $ \mathbf{b}+3\mathbf{c} $ is collinear with a, then ( $ \lambda $ being some non-zero scalar) $ \mathbf{a}+2\mathbf{b}+6\mathbf{c} $ is equal to
[AIEEE 2004]
Options:
A) $ \lambda \mathbf{a} $
B) $ \lambda \mathbf{b} $
C) $ \lambda \mathbf{c} $
D) 0
Show Answer
Answer:
Correct Answer: D
Solution:
- Let $ \mathbf{a}+2\mathbf{b}=x\mathbf{c} $ and $ \mathbf{b}+3\mathbf{c}=y\mathbf{a}, $ then $ \mathbf{a}+2\mathbf{b}+6\mathbf{c}=(x+6)\mathbf{c} $ and $ \mathbf{a}+2\mathbf{b}+6\mathbf{c}=(1+2y)\mathbf{a} $ So, $ (x+6)\mathbf{c}=(1+2y)\mathbf{a} $ Since $ \mathbf{a} $ and $ \mathbf{c} $ are non-zero and non-collinear, we have $ x+6=0 $ and $ 1+2y=0 $ i.e., $ x=-6 $ and $ y=-\frac{1}{2}. $ In either case, we have $ \mathbf{a}+2\mathbf{b}+6\mathbf{c}=\mathbf{0} $ .
BETA
