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(I) For any vector $\vec { \mathbf { v } } = V _ { x } \hat { \mathbf { i } } + V _ { y } \hat { \mathbf { j } } + V _ { z } \hat { \mathbf { k } }$ show that $V _ { x } = \hat { \mathbf { i } } \cdot \vec { \mathbf { v } } , \quad V _ { y } = \hat { \mathbf { j } } \cdot \vec { \mathbf { v } } , \quad V _ { z } = \hat { \mathbf { k } } \cdot \vec { \mathbf { v } }$

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$$\begin{array}{l}{\vec{V} \bullet \hat{i}=V_{x}(1)+V_{y}(0)+V_{z}(0)=V_{x}} \\ {\vec{V} \bullet \hat{j}=V_{x}(0)+V_{y}(1)+V_{z}(0)=V_{y}} \\ {\vec{V} \bullet \hat{k}=V_{x}(0)+V_{y}(0)+V_{z}(1)=V_{z}}\end{array}$$

Physics 101 Mechanics

Chapter 7

Work and Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Hope College

McMaster University

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.

05:06

For any vector $\overright…

07:13

Prove: For any vectors $u,…

02:23

Show that if $\vec{u}$ and…

01:53

Let $\boldsymbol{u}=\langl…

02:04

(a) If $\mathbf{u} \cdot \…

00:54

01:01

01:25

00:59

01:23

7.17. So we want to show that for any vector that has, you know, an x y and Z components that the dot product of the corresponding unit vector is the bat component. So say, for example, the cake I had not be. We'll have a visa, Becks times. I had not. I have. You know, this is one plus visa. Why, uh, well, it's community is it doesn't matter what order you write it up. But I had not jihad. This is zero and the subsea. I had not k hat. This is also zero, the sort of by the definition of how the unit vectors were. So this is Visa, Becks. So for J hat, you just basically go through and do the same thing over again. Not a product of jail. I had a zero because there are perpendicular to each other. The dot product of J hat with itself is one. And then jay kay hat also haven't dot product zero because they're again perpendicular. So this is Visa Boy. And so for K hat, we have visa next times. Okay, hat. But I had zero v supply. Kay had not J had also zero plus v sub z times k had not kay had. This is one. And so we see that the dot product of a, uh basis vector or a unit vector whatever Ah, with given vector of interest gives the corresponding component as you'd expect.

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