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Question
If `veca` and `vecb` are two collinear vectors, then which of the following are incorrect:
Options
`vecb = λveca`, for some scalar λ
`veca = pm vecb`
The respective components of `veca` and `vecb` are not proportional.
Both the vectors `veca` and `vecb` have the same direction but different magnitudes.
Solution
Both the vectors `veca` and `vecb` have the same direction but different magnitudes.
Explanation:
If `veca and vecb` are two collinear vectors, then they are parallel.
Therefore, we have:
`vecb = lambdaveca` (For some scalar λ)
If λ = ±1, a = ±b
If `veca = a_1hati + a_2hatj + a_3hatk and vecb = b_1hati + b_2hatj + b_3hatk`, then
`vecb = lambdaveca`
⇒ `b_1hati + b_2hatj + b_3hatk = lambda(a_1hati + a_2hatj + a_3hatk)`
⇒ `b_1hati + b_2hatj + b_3hatk = (lambdaa_1)hati + (lambdaa_2)hatj + (lambdaa_3)hatk`
⇒ `b_1 = lambdaa_1, b_2 = lambdaa_2, b_3 = lambdaa_3`
⇒ `b_1/a_1 = b_2/a_2 = b_3/a_3 = lambda`
Thus, the respective components of `veca and vecb` are proportional.
However, vectors `veca and vecb` can have different directions.
Hence, the statement given in D is incorrect.
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