a. 1. The most basic difference is whether you are looking for a scalar result or a Vector result. Dot product gives you a scalar result whereas Cross product gives you a Vector result.
2. Basically the purpose of application differs. Dot product gives you an expression for Work done whereas Cross product gives you an expression for Torque.
3. Dot product would have been maximum if both the vectors had acted in the same direction, but a Cross product = 0 if both the vectors are in the same direction.
More detailed explanation is given below-
A. Dot product is simply the effect of one vector on the other should it be acting in the same direction as the base vector. Assume two vectors acting on a mass in the exact opposite directions. They would nullify each other. But if they were acting on the body in the same direction, the effect of the forces would be maximum.
Assume vector a = 2i and vector b = 4i (both are acting along i direction). The dot product = 2i . 4i = 8 (i.i = 1). Whereas dot product of 21 with 4j = 0 (no net resultant if both the vectors are in different directions).
B. Consider the realistic example below (For Dot product).
Whenever someone talks about dot product, remember the example of work done.
Work is the product of force and displacement should the force be acting in the same direction as the displacement.
But if the force was acting along some other direction other than the direction of displacement, then the entire force does not cause displacement. There is only a percentage of this force that goes into causing this displacement. And this percentage of force will be lesser than the actual force. Thus, we consider the cosine of the force (max value of cosine = 1 and hence this would result in a percentage of force much smaller than the actual magnitude of the force).
Thus, work done = F.s = F. s. cos $
If the force was acting in the same direction as the resulting displacement, then the magnitude of the work done is maximum (as cos 0 = 1). Dot product is simply the effect of one vector on the other should it be acting in the same direction as the base vector (in this case Force vector and the displacement vector).
In the example above, the person is pushing the object in the same direction as the displacement and hence this would result in the maximum work done by the person. Hence with a smaller force applied, he can do the maximum work that is possible and hence he needs lesser effort to push the object along the direction of displacement. But if he was pushing at an inclined angle, then he would have had to apply more effort as the work exerted by him would be low.
See the image above. The person is applying a force that is inclined. Hence he needs to apply more force.
So basically this is where Dot product Come into play.
In the first example, basically the forces are in the same direction. But in the second example, the forces are not in the same direction (one vector is projected over the other). Dot product gives the projection of one vector over another.
C. Cross product on the other side is going to give you a Vector which will be at right angle to both the vectors between which you establish Cross products. So if you can form a plane containing both a and b vectors, then the cross product will point in a direction perpendicular to the plane.
In Finite Element Analysis, if you are establish a Co-ordinate system or a normal to a plane or a Surface, then Cross product is what you should be looking at.
D. A dot product would have been maximum when both the vectors had acted in the same direction, but a Cross product = 0 if both the vectors are in the same direction.
Usage
Cross product finds it’s usage in Torque. Torque = Magnitude of Force * Distance from axis of rotation. As distance is more, the Torque is also more. This is why we need to open doors by applying force as far as possible from the door hinges. The hinges of the door are the axis of rotation.
Torque is the cross product between Force and the distance vector (T = r * F * sin (Theta)). Thus the angle between the Force vector and the distance vector is also accounted for.
When we open a door, we apply a Force which is 90 degrees to the distance vector. Sin 90 = 1 and this is why we open the door by applying a force perpendicular to the distance vector. As this will generate the maximum Torque.
In the above figure,
a. In figure a, The force is applied exactly at the position of the rotational axis, thus the distance = 0. Thus Torque = F * r = 0. Irrespective of the force applied, Torque = 0.
b. In figure b, The force is applied at the maximum possible distance away from the rotational axis. Thus, we get very high Torque.
c. As the point of force application gets more closer to the rotational axis, there is lesser Torque.
d. We saw that Torque also depends upon the angle between the Force and the Distance vectors. Here the force is parallel to the distance vector. Thus, angle = 0 and sin (0) = 0. Thus, there is no Torque.
Thus for maximum torque, the force must be applied perpendicular to the distance vector.
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