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(II) Prove that $\vec { \mathbf { A } } \cdot \vec { \mathbf { B } } = A _ { x } B _ { x } + A _ { y } B _ { y } + A _ { z } B _ { z }$ starting from Eq. 2 and using the distributive property $( \vec { \mathbf { A } } \cdot \vec { \mathbf { B } } = A B \cos \theta ,$ proved in Problem 33$)$

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$A_{x} B_{x}+A_{y} B_{y}+A_{z} B_{z}$

Physics 101 Mechanics

Chapter 7

Work and Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Cornell University

Hope College

University of Sheffield

Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

03:47

In physics, the kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest. The kinetic energy of a rotating object is the sum of the kinetic energies of the object's parts.

06:50

In $3-26,$ prove that each…

01:06

Prove that:If $\cos ^{…

05:50

$$\text { If } \sin \t…

02:13

If $\cos ^{-1}\left(\frac{…

01:04

If $y=A \sin x+B \cos x$, …

01:01

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01:25

03:52

01:35

so here we're actually trying to prove the dot formula. So let's say that I had times the dark product between I had and I add will equal one. And this will equal thank you. The dark product between J had and J hod and end the dark part of between K hat and Cape Any other combinations. So I j this will equal j Okay, this will equal high. Okay, this'll equal k i this week k j And this legal of course J I and this legal zero. So given this, we can say that the not product between two vectors, by definition, would be simply ace of Ex ID plus a. So why, Hey, plus any sub Z K hat. And then times rather did our product between A and B X I had plus B y j hat was easy. Okay, At this point, we can say that this is going to be equal to x The X times one plus a x b Why time? Zero plus a xB ze time zero plus a y be wide times one plus a y times p x time zero plus a wide times peasy time zero and then plus a cz times B z times one ese d x times zero plus ese B Y times zero. So essentially all of them will cancel out. And after everything is cancelled out, we're still simply left at a That product between Victor and Victor B will be equal to xB X plus y b. Why first, ese. You see? So this would be proving that, uh dot product formula that is the end of the solution. Thank you for watching.

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