What type of triangle is made? Using the value of R and R y, how would you calculate Rx? Verify the value of Rx by calculating it on your own. A right triangle is made. Create another vector that is not completely horizontal or completely vertical. Drag it so that its tail is touching the tip of the vector from 9. What are the values of Rx? Add the vector from 11 to your picture.
Move the new vector so that its tail is at 0,0. Draw this vector to scale on the drawing you made in 9 and For your newest vector, what are the values of Rx? These values make sense since when if you add the component X of the first vector and the second vector the resulting value of Rx is If you add the Y component of the first vector and the second vector the resulting value of Ry is The arc tan of Rx and Ry component of the vector is The manual computation of the R is For your newest vector, how was Rx determined?
How was Ry determined? How was R determined? Try to verify the results the computer gave you by calculating the R yourself. In general, if you know the measurements of the horizontal parts components of vectors and you know the measurements of the vertical components of vectors, how can you find their sum resultant?
To find their sum resultant, the first things that needs to be done is to add the X components of the vectors then add also the Y components of the vectors to get Rx and Ry. Use the simulation to represent your path. Draw and label your vector on your lab write-up. How far from where you started did you end up?
From the figure above we can see that the sum vector has the magnitude of What is this property called? Does this property hold for vector addition? Try it with the scenario above by first walking 10 steps west and then walking 20 steps north. Compare the result to the sum in part b.
Commutative property of addition. This property states that when two numbers are added, the sum is the same regardless of the grouping of the addends.
This property holds on the vector addition. The sum in part a and part b are the same. You are going for a drive, but a detour takes you out of the way of your destination. You drive north for 10 miles, turn right and drive east for 5 miles, turn to the north and drive for 3 miles, drive west for 10 miles, and arrive at your destination.
Ended up What direction would a compass read? If you want to travel to your destination without following the street you should take the angle degrees headed northwest. For example, start with the 5 miles east, then 3 miles north, etc.
Compare the result to the sum in part a. Draw the vectors in the order you used. As shown below the resultant in part a is the same with the resultant in part b. The wind is blowing at 4 mph from due north. Now what is your airspeed and what direction are you flying? If your destination is to the northeast, how would you change your speed or direction so you might make it there? Test your answer using the sim. A baseball weighing 0. Label the vectors on your diagram.
You hike for 1. What would a compass read? The results are then more components, which then have to be reconstructed into a vector. So we can use perpendicular coordinate systems to describe vectors in terms of their components. Essentially this means that to describe a vector in terms of a set of three axes, we need to know three numbers.
Vectors have magnitude and direction, and with unit vectors we can mathematically break up the vector into those two parts. The magnitude is just a number with physical units without direction, and a unit vector is a vector without units that has a length of 1, so that it can be scaled to any length without contributing anything to the magnitude.
Therefore we can write a vector as a simple product:. The diagram below gives a graphic description of how this construction works for a few common physical vectors. The unit vectors provide a very basic template by defining the direction, and the magnitude fills in the template by contributing the girth and 'flavor' physical units of the vector.
If we combine this notion with components, we can write any vector as a sum of components multiplying unit vectors in the directions of the three spatial dimensions. So specifically, we have:. Now we can just use this as a mathematical representation of vectors, and we do not have to appeal to geometry at all. For example,. Repeat the calculation of Example 1. This matches the answer found in example 1. Definition of a Vector Just being able to put numbers on physical quantities is not sufficient for describing nature.
The system conforms with what you will find in general physics texts, with some minor differences. One difference is that general physics texts often don't have a symbol for "distance traveled". They just don't calculate the distance traveled and therefore don't need a symbol.
Also, although books careful ones are coming out now with a vector notation that uses letters in boldface with an arrow on top, many books still use only boldface. My applets and text use boldface plus the arrow. The reason the arrow is important, I think, is because that is what students will use in their handwriting. Students have a great tendency to omit the arrow, and so we must model good practice for them.
Anyway, the arrow is used also in the Alberta Education manuals. So the arrow is not an issue. I just would like to point out that not all textbooks use arrows and that this is one difference between the notation proposed here and that found in general physics texts. I can recommend it. Displacement: or D Note : the notation " D " is hardly ever applicable; one is usually not interested in the change of a displacement; one is interested in displacement, which is a change in position.
On the Alberta Learning Data Sheets from , displacement is denoted by " d " with an arrow on top, but no boldface. I don't know if that notation has changed. Assuming it is still the notation used now for displacement, then " D ", with either boldface or plainface " d ", should not come up in any instances that I can think of.
Also, students tend to refer to g as "gravity". This usage should be discouraged. There is an "acceleration due to gravity", a "force due to gravity", a "potential energy due to gravity", etc. They should not all be referred to by the one word "gravity". The following notational distinctions are a bit tedious at times, but consistent and unambiguous. They are intended to distinguish between components and magnitudes when everything is in one dimension only. The following applies to motion along an x -axis.
Velocity component: v x Speed: v Note : It is common to write v instead of " v x "; this notation is ambiguous because " v " and " v x " are not the same; the speed v is a positive quantity while the velocity component v x can be both positive and negative. In a given context, v x may always be positive; in this case, one can safely write " v " instead of " v x "; however, it would be good to point out this change in notation.
Even if v x may be negative in a given context, one may want to write " v " instead of " v x ", to use a less cumbersome notation.
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