Two forces act on an object: 𝐅 one equals 20 newtons 𝐢 hat plus 50 newtons 𝐣 hat and 𝐅 two equals negative 30 newtons 𝐢 hat plus 10 newtons 𝐣 hat. What is the total force acting on the object in component form?
So in this question, we’re given two forces represented by the vectors 𝐅 one and 𝐅 two, which are given to us in component form. We’re told that these two forces act on an object, and we’re asked to work out what the total force acting on that object is. Now whenever we have multiple forces acting on the same object, the total force acting on that object is given by the resultant of all of those individual forces. So in order to find this total force, what we need to do is find the resultant of the two vectors 𝐅 one and 𝐅 two.
We can recall that the resultant of two vectors is defined as the sum of those two vectors and that in order to add vectors together, we need to add together the 𝑥-components and the 𝑦-components of those vectors separately. So now let’s add together our force vectors 𝐅 one and 𝐅 two from the question. If we recall that 𝐢 hat is the unit vector in the 𝑥-direction and 𝐣 hat is the unit vector in the 𝑦-direction, then we can identify the 𝑥-component of 𝐅 one as 20 newtons and the 𝑦-component of 𝐅 one as 50 newtons. And similarly, for 𝐅 two, we can identify the 𝑥-component as negative 30 newtons and the 𝑦-component as 10 newtons.
So if we calculate the sum 𝐅 one plus 𝐅 two, then first we need the sum of the 𝑥-components, which is 20 newtons plus negative 30 newtons. And since this is the 𝑥-component, we need to multiply this by the unit vector 𝐢 hat. Then we need the sum of the 𝑦-components, which is 50 newtons plus 10 newtons. And since this is the 𝑦-component of our resultant vector, we need to multiply it by the unit vector 𝐣 hat.
If we then evaluate the sums in each of the components, for the 𝑥-component we have 20 newtons plus negative 30 newtons, which gives us negative 10 newtons, and for the 𝑦-component we have 50 newtons plus 10 newtons, which gives us 60 newtons. So we have that the sum 𝐅 one plus 𝐅 two is equal to negative 10 newtons 𝐢 hat plus 60 newtons 𝐣 hat.
Now recall that at the beginning of the question we said that the sum of two vectors gave the resultant of those two vectors. And since the total force acting on an object is equal to the resultant of all the individual forces, then this sum that we have calculated, 𝐅 one plus 𝐅 two, gives us the total force. And we have our answer that the total force acting on the object in component form is equal to negative 10 newtons 𝐢 hat plus 60 newtons 𝐣 hat.