Sometimes, we will get situations where we need to calculate the angle between 2 forces,
or 2 sides on an object in 3D space. In that case, using the dot product can be really helpful.
So if we have 2 vectors with an angle, we can write the dot product like this.
If you’re trying to find the angle, we can write the equation like this to calculate the angle
between the 2 sides. In another situation, we might need to find the components of a vector
that is parallel and perpendicular to a line. The parallel component is sometimes shown like
this, or like this, and its referred to as the projection of a force. To find it, all we do is
take the dot product between the force expressed in cartesian form and the unit vector of the axis,
or side where the projection is supposed to be. If we need to find the perpendicular component,
then we can use this equation to do it, which is just the Pythagorean theorem.
Here, A is the magnitude of the force, this is the parallel component and this is the perpendicular
component. This will become much clearer with examples, but to do so, you should be able to
express forces in cartesian form, and be able to write position vectors along with unit vectors.
In the previous video, I go through this step by step, so if you need a refresh,
or you forgot, please check the description. Now let’s move on to some examples, and see
how we can find angles, projections and more. Let’s take a look at this question where all we
need to do is figure out the angle between the sides of the triangular plate. We are going to
do it using the dot product. So let’s write down the equation to figure out an angle.
So first, we need 2 position vectors, for the 2 sides of the triangle. One from A to B and one
from A to C. Let’s write down the locations of points A, B and C (pause after each letter). Now
we can write our first position vector. Let’s simplify. Next, a position vector from A to C.
Now, we need to find the magnitude of these position vectors. Pause 2 seconds.
Next, we will take the dot product between the 2 position vectors. So all we’re doing
is multiplying I components by I components, j components by j components and k components by k
components. The result is a scalar value. Now all that is left is to plug everything
into our equation. Solving gives us our answer. Let’s take a look at this question involving
projections. We need to figure out the magnitudes of the projected components of the force along
cables AB and AC. When we have questions asking for projections, we need to find unit vectors.
So in this problem, we need a unit vector for cable AB and cable AC. So let’s start
by writing down the locations of each point. So we have points A, B and C. Next, we need
position vectors from A to B and A to C. Now we can find the magnitude. Okay, so finally, we can
divide each component in our position vector by the magnitude, and we get our unit vectors.
We are now ready to do the dot product. First, the projection of force F onto cable AB. For that,
all we do is take the dot product between force F and the unit vector AB. Remember, force F
has to be expressed in cartesian form, and since the question already gives it in cartesian form,
we can just take the dot product. Since we are asked for the magnitude, remember,
it’ll be positive. The negative sign means that our components points in the opposite direction
to the unit vector. Next, the projection of force F onto cable AC. Those are our answers.
Let’s take a look at this problem where we need to find the projected component of the 600 N force
along rod AC. Unlike the previous question, our force isn’t given in cartesian form,
so we need to express it as such. After that, we need a unit vector for rod AC.
So do to all this, we first need to write down the locations of all the points of interest.
First, point A. Next, we need to figure out point B. The question says point B is at
the midpoint of rod AC. So the x component would be 1.5m in the negative x direction,
the y component would be 2m and the zed component would also be 2m. We are just dividing the
distances by 2. Next, we have points C and D. To express the 600 N force in cartesian form,
we need a position vector from B to D. Now we need the magnitude of this vector.
Next step is to divide each component in our position vector by the magnitude,
which gives us our unit vector. Lastly, we just multiply the magnitude
of the force by the unit vector, and that is our force expressed in cartesian form.
Now we need a unit vector for rod AC. The process
is pretty much the same. First, a position vector from A to C.
Now the magnitude. And lastly, the unit vector.
Let’s write it in decimal form. Now all that’s left is to take the
dot product between the force and the unit vector. Solving this gives us the component
of the force that is along rod AC. Now we need the perpendicular component. For that,
we use this equation. So F is the magnitude of our Force, so it’s 600 N. The parallel component is
what we just found, so now, we can solve for the perpendicular component. Those are our answers.
That should cover the types of problems you will face when it comes to dot products and forces.
Thanks for watching and best of luck with your studies!