Which of the following is NOT correct if we have \(\vec{a}\) and \(\vec{b}\) as two collinear vectors?

Explanation:

• Collinear vectors are those vectors that are having the same line of actions

\(\vec{b} = λ \vec{a}\), λ ϵ R 

• They may have different magnitudes and may also be opposite in direction to one another but they must be parallel to the same line.

For, λ  = ± 1,

\(⇒ \vec{b} = \pm \vec{a}\)

So, the statement (iii) and (iv) are correct.

• Now, let,

\(\vec a = p\hat i + q\hat j+r\hat k\) and \(\vec b = x\hat i + y\hat j+z\hat k\)

• Now, using \(\vec{b} = λ \vec{a}\),

\(x\hat i + y\hat j+z\hat k = \lambda (p\hat i + q\hat j+r\hat k)\)

• From the above, we can say that the elements of the vector are proportional so statement (i) is correct.
• The statement (ii) is false as they may have different or same magnitudes and may be opposite in direction.

Note: There were two statements that are incorrect for this question. The statement (i) was like "Elements of both vectors are not proportional" initially which is modified to get the one answer for this question.