Work as dot product and its implications

[sc4s2cg]

I am reading Kaplan physics Ch. 2 and it states:

"You'll notice that work is a dot product; as such, it is a function of the cosine of the angle between the vectors" - I understand this so far since dot product of A * B = absA * absB * costheta

"This also means that only forces (or components of forces) parallel or antiparallel to the displacement vector will do work (that is, transfer energy)."

I don't understand the logic behind this. Granted, all I know about dot/cross product is their equation and relevance to scalars v vectors. Conceptually, how does it follow that since work is a dot product, only forces parallel or antiparallel deltaX will transfer energy?

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[chemist16]

The easiest way to do this is to imagine two vectors, A and B. Draw them so that their tails touch and there is an angle theta between them. Now, let's step back for a second and discuss the physical principle behind work. If you're pushing something along the ground horizontally and apply a force that is downward and to the right (-y and +x directions), only the rightward force will produce a displacement and consequently, work. Therefore, the force you apply in the downward direction is irrelevant to the problem - no matter how hard you push in the -y direction, the thing isn't going to move in that direction and you're not going to do work on it. In simpler terms, an applied force does work if and only if it produces a displacement in the direction of the applied force.

Dot product accomplishes exactly this. Say vector A is your applied force and vector B is displacement. Only the component of the force in the direction of displacement produces work. So if I were to ask you what that component is, you could solve it by making a right triangle with A being the hypotenuse and the angle being theta, as noted above. So what's the component of A in the direction of displacement, namely the direction of vector B? Well, it's A*cos(theta). So then, work is applied force in the direction of displacement times displacement, or B*A*cos(theta). In other words, it's the dot product!

Now, a more intuitive explanation. A dot product literally projects one vector onto the same direction as some other vector - in this case, it projects A onto B. If you can see how it projects A directly onto B, you will see immediately why only the parallel or antiparallel component of force will transfer energy. Any component of the force in the y-direction isn't projected onto B. For example, take a vector A that is perpendicular to vector B. You can immediately tell that there is no component of A in the direction of B. This would be like pushing straight down on a box and expecting it to move sideways. So, no work should be done. The dot product naturally confirms this, since it's just a projection of A onto B. A*B*cos(pi/2) = 0.

 

[sc4s2cg]

Wow thank you, this explanation hits right where it needs to. I have one question left: since dot product is about working with scalars (at least according to Kaplan), how come we are taking the direction (vector) into account as well?

Would you mind explaining cross-product in a similar fashion? As before, my conceptual knowledge only extends to absA*absB*sintheta = A x B, but this time vectors instead of scalars.

 

[chemist16]

You're not taking direction into account per se. So a dot product is single-dimensional. By projecting one vector onto another, you're saying "no direction other than this one matters" - in other words, only one dimension matters (namely the direction of the second vector).

Cross product is similar. The best way to describe this is using torque. As you know, for torque, only the force applied in a direction perpendicular to the lever arm results in torque. Imagine a door handle. You can push down on it to turn it and open the door. But if you try pulling or pushing on it horizontally, it's not going to move. It's just like any lever. So, in other words, only the component of force that is perpendicular to the lever arm causes a torque. So, what's that component? Draw a lever arm and an applied force. Say the door handle is facing right (pivot point is on the left) and you're applying a force at the right end of the lever. Your force is into the door handle and downward, or -x and -y directions. From torque, you know that only the -y component of the force matters. So draw your vectors again, tails touching with an angle theta between them. On a coordinate axis, it should look like one horizontal vector A and another one B pointing in the third quadrant. As I said, only the y-component of B matters. What's that y-component? Well, it's simply B*sin(theta). And torque is perpendicularly applied force times the length of the lever arm, or A*B*sin(theta).

So intuitively, the cross product is the opposite of projecting one vector onto another. It's taking only the perpendicular component of one vector with respect to another.

Now, recall that your cross product is a vector with a certain direction. That direction is conventionally given by the right-hand rule.