[bobby19]

Hey,

I am able to do numerical questions involving vector components and
Vector projections, but I do not fully understand the geometric
Significance behind them. What are the differences between the two?
What exactly is meant when asked find the prjoection of vector a
Onto b?

[Capuchin]

The projection of one vector onto another is the dot product of the vectors. As you rightly say it gives you the length of one vector along the other.

This is of huge importance in physics. For example if you have a force in one direction, but you need the force felt in a different direction, then you can use the dot product to find it.

When you take a force and work out the x and y components of it, you are doing the dot product of the force with the unit vectors I and j which have length 1, so they aren't expressed in the calculation you do, to give you a force along the x and y direction.

As another example, Work = Force . Displacement
This produces a SCALAR quantity (work) from the 2 VECTORS Force and Displacement.

When you first learn this equation, the force is in the same direction as the displacement, so cos(theta) is 1, but if it's a 2 dimensional problem you need the dot product to give you the true result.

I hope this helps a little.