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The difference between scalar and vector quantities is very simple. Vectors tell you information about direction, scalars don't. An example of a scalar quantity is the amount of money you have. 37 cents doesn't give a direction. It's not 37 cents up or 37 cents east, that doesn't make any sense. (Cents? Sorry, couldn't resist the pun). Driving down the street, however, is a vector quantity. There is a number, 4m/s, and there is a direction, north, for example. Most of the time the direction is indicated by a sign (-9.8m/s2 means acceleration downward) or a degree (10m/s, 41° North of East).
Here are some examples of scalar quantities:
Temperature
Mass
Speed
Here are some examples of vector quantities:
Velocity
Acceleration
Weight (because weight is really a force downward)
We represent vectors with arrows. The length of a vector is called it's magnitude. The magnitude is just the number part. In the vector is 123 m/s North, the magnitude is 123 m/s and the direction is North. The vector should also be drawn in its direction, or a representation of its direction. What the hell am I talking about? Just draw North arrows pointing up, west arrows pointing left, you get the idea. Remember, if you change the direction that a vector points, it is no longer the same vector! Vectors depend on both their magnitude (length) and direction.
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These are the easiest kind of vectors to add. You can just pretend the directions aren't there and add the numbers. For example if you have to add 17m, East, and 13m East, just add the numbers and slap the direction on to the end. 30m, East is the correct answer. Another Example: 5m/s, 30° North of East + 2m/s, 30° North of East=7m/s, 30° North of East. It's pretty simple.
This one is a little trickier. What do you do when you have to add 10m, North and 30m, South? Step one: Find the larger number, in this case 30. Now subtract the smaller number from that. 30 - 10=20. Now just attach the direction from the larger number. 20m, South is the answer. It's just like adding negative numbers in math. Here is another example. 700N, 0° North of East + 300N, 0° North of West=400N, 0° North of East. As a side note, if you are asked to add many vectors, (I.e.: 1m, East + 2m, East + 3m, West + 4m, West) First add all the vectors pointing in the same direction (so you are left with 3m, East + 7m, West) then follow the steps above. (Final result equals 4m, West).
Now that we can add vectors on the same line, it's time to move onto the next section. Perpendicular Vectors. Your teacher asks you to add 300m, 0° North of East and 400m 90° North of East. Here's how you do it. Remember Geometry? Remember the Pythagorean theorem? (a2 + b2 =c2) We have to use that. Since the vectors are at right angles to each other, we can form a right triangle out of them. The resultant vector (the answer) will be the third side of the triangle. Use the vectors you have as the bases of the triangle and plug it into the equation.
"Yeah! We have the answer, now we can go out drinking right?" Not quite. We still need to find the exact angle that the vector points. To do that we need to sell our soul to the demon of trigonometry. If you remember Tan=Opposite side/adjacent side. So, Tan=400m/300m, Tan=1.3333, Inverse Tan=53°. So the final answer is 500m, 53° North of East.
Vector Components
Vector components are little vectors that add up to the vector you are working with. For example, If you took a vector like, 40m/s, West, its components could be 20m/s, West and 20m/s, West, or 60m/s, West, and 20m/s, East. Any vectors that add up to the one you are working with are components. Any vector can be broken down into an infinite number of components. Luckily, in most situations we only care about two.
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Perpendicular Components
Perpendicular components are the most useful types. They are two vectors that are perpendicular to each other and add up to the vector you are working with. To find the perpendicular components of a vector we need to use trigonometry. Lets do an example.
Given a vector, 175N, 50° North of East, find the perpendicular components. To find the vertical component, we use the SIN of 50°, .766. We multiply that by the length of 175N and get a vertical component of 134N, North. To find the horizontal component we take the COS of 50°, .642 and multiply it by 175N to get a horizontal component of 112N, East. The reason we are able to do this is because the perpendicular components form the bases of a right triangle with the resultant vector (the vector you are working with) as the hypotenuse.
| In the animated gif on the right, the red vector is the result of the addition of the two blue/gray vectors. The blue/gray vectors are the perpendicular components of the red vector. |
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If any of the above was unclear, or if you have any comments or suggestions, please E-mail me!
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