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Question: Which of the following statements are correct: If M is the midpoint of AB and O is any point then ...

Which of the following statements are correct:
If M is the midpoint of AB and O is any point then
(This question has multiple correct options)

(a) O⃗M⃗=O⃗A⃗+M⃗A⃗ (b) O⃗M⃗=O⃗A⃗−M⃗A⃗ (c) O⃗M⃗=12(O⃗A⃗−O⃗B⃗) (d) O⃗M⃗=12(O⃗B⃗+O⃗A⃗)  (a){\text{ }}\vec O\vec M = \vec O\vec A + \vec M\vec A \\\ (b){\text{ }}\vec O\vec M = \vec O\vec A - \vec M\vec A \\\ (c){\text{ }}\vec O\vec M = \dfrac{1}{2}\left( {\vec O\vec A - \vec O\vec B} \right) \\\ (d){\text{ }}\vec O\vec M = \dfrac{1}{2}\left( {\vec O\vec B + \vec O\vec A} \right) \\\
Explanation

Solution

Hint: In this question it has been given that M is the midpoint of AB and O is any point. Use the concept of vectors that if a vector is reversed its sign gets reversed as well that is A⃗M⃗=−M⃗A⃗\vec A\vec M = - \vec M\vec A along with the concepts of mid-point that a mid-points divides the line into two equal portions to get the answer.

Complete step-by-step answer:

It is given M is the midpoint of line AB.
⇒A⃗M⃗=M⃗B⃗\Rightarrow \vec A\vec M = \vec M\vec B………………………. (1)
Now O is any point.
Now position vector OM is written as
⇒ O⃗M⃗=O⃗A⃗+A⃗M⃗\Rightarrow {\text{ }}\vec O\vec M = \vec O\vec A + \vec A\vec M…………………… (2)
And we all know if the direction of the vector is changed then the magnitude of the vector is changed (i.e. if magnitude is positive it becomes negative or vice-versa).
⇒A⃗M⃗=−M⃗A⃗\Rightarrow \vec A\vec M = - \vec M\vec A
Substitute this value in equation (2) we have,
⇒ O⃗M⃗=O⃗A⃗−M⃗A⃗\Rightarrow {\text{ }}\vec O\vec M = \vec O\vec A - \vec M\vec A (Which is option (b))
Now from equation (1)
⇒A⃗M⃗=M⃗B⃗\Rightarrow \vec A\vec M = \vec M\vec B
Now break the vectors in terms of position vector of O we have,
⇒A⃗O⃗+O⃗M⃗=M⃗O⃗+O⃗B⃗\Rightarrow \vec A\vec O + \vec O\vec M = \vec M\vec O + \vec O\vec B
Now according to above property we have
⇒−O⃗A⃗+O⃗M⃗=−O⃗M⃗+O⃗B⃗\Rightarrow - \vec O\vec A + \vec O\vec M = - \vec O\vec M + \vec O\vec B
Now simplify the above equation we have,
⇒2O⃗M⃗=O⃗A⃗+O⃗B⃗\Rightarrow 2\vec O\vec M = \vec O\vec A + \vec O\vec B
⇒O⃗M⃗=12(O⃗A⃗+O⃗B⃗)\Rightarrow \vec O\vec M = \dfrac{1}{2}\left( {\vec O\vec A + \vec O\vec B} \right) (Which is option number (d))
Hence options (b) and (d) are correct.

Note – Whenever we face such types of problems the key concept is simply to have a good understanding of vectors. The diagrammatic representation of the information given in question helps in better understanding of the question. The midpoint properties will eventually take you on the right track to get the answer.