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.

Answer 1

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.