Question

Help please!!!!(:
A wall clock has a minute hand with a length of 0.56 m and an hour hand with a length of 0.23 m. Take the center of the clock as the origin, and use a Cartesian coordinate system with the positive x axis pointing to 3 o'clock and the positive y axis pointing to 12 o'clock. Write the vector that describes the displacement of a fly if it quickly goes from the tip of the minute hand to the tip of the hour hand at 3:00 P.M. (Let Dvec represents the displacement of the fly.)

Vector D=_____m(i-hat)+_____m(j-hat)

Answer 1

@ganeshie8

Answer 2

Hi! So it looks like you want the answer in the for of \(\vec D = a\hat i+b\hat j\).

How familiar are you with vectors?

Answer 3

@pdd21

Answer 4

yes! and how to do it lol @theEric

Answer 5

i'm somewhat familiar with vectors.

Answer 6

Okay! Well, that's a good start! :)

You have a coordinate system laid out by the clock, where the \(x\) and \(y\) axes run through the center, where the \(y\)-axis goes up to the 12 spot and \(x\)-axis goes right to the 3 spot.

Drawing a picture is usually helpful!

Answer 7

There is your coordinate system!

Answer 8

Cartesian just means we're talking about \(x\) and \(y\) rectangular stuff.

Answer 9

Now we want to look at the hour and minute hands - to continue building up this picture. I will draw \(\sf vectors\) that point to the \(\sf ends\) of the hands, because that describes the two \(\sf positions\) of the fly. That is important, because the \(\sf displacement\) you seek is the \(\sf change\ in\ position\), from the end of the minute hand to the end of the hour hand.

Answer 10

okay

Answer 11

i'm following so far(:

Answer 12

Those vectors point to the \(\sf position\) of the ends of the hands! The hour hand is the shorter one, pointed at three. There are no extra minutes, so the minute hand is at 12!

Answer 13

I'm glad! Thanks for letting me know! :D

Answer 14

Now, I'm guessing you are used to expressing vectors with \(\hat i\) and \(\hat j\) - unit vector notation. So, what would the vector to the end of the minute hand be? What about the vector to the end of the hour hand?

Answer 15

(I'll give you hints if you need.)

Answer 16

so i-hat would be the x-axis right? and j-hat is the y-axis. the end of the hour hand is 3? and the end of the min hand is 12?

Answer 17

Right, right, right, and right.

How it works, then, is that \(\hat i\) is a unit vector in the \(x\) direction. When you multiply anything to it, say \(a\), the result is a new vector, where the length is changed based on \(a\)! So, adjusting \(a\), you can have any length of the \(x\) direction vector. That covers your \(x\) component.

Then, for your vector's \(y\) component, you have your \(\hat j\), which works the same way but in the \(y\) direction.

Sometimes you combine these components with "vector addition," which makes the result equal a vector that has both \(x\) and \(y\) components, and so it looks slanted!

Now, in your case, these two hands line up with the axes, so you don't have two combined components for either of them.