how do we express newtons second law in vector form

Dear Student!

This is well explained in the study material on our website. Here's a chunk.

Newton’s Second Law of Motion

• Statement

The rate of change of linear momentum of a body is directly proportional to the external force applied on the body, and this change takes place always in the direction of force applied.

Let,

m = Mass of the body

= Velocity of the body

Therefore, the linear momentum of the body is,

Let,

= External force

 = Change in linear momentum

According to Newton’s second law,

Where, k is the constant of proportionality

Taking the limit becomes 

In S.I, k = 1

∴

Thus, the second law of motion gives us the measure of force.