Precalculus (2018 edition)
Watch Sal figure out which of a few options could be represented by a vector. Remember that vectors have both magnitude and direction! Created by Sal Khan.
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- I don't see how the point (5,5) by itself could represent a vector since they didn't say it was the endpoint of the vector nor did they say the vector started at the origin.(108 votes)
- I agree that a point is not a vector; it has neither direction nor magnitude. If you want to say that there is an implied direction and magnitude from the origin, then I think by the same argument you could say that 5 and -5 are vectors of the same magnitude but opposite directions when they are placed on a number line and measured from the origin (zero).
 I think the confusion is about what (5,5) represents. Is it a point, or the location of a point? When I think of the pair of numbers as a Cartesian coordinate, it becomes clear that these two numbers represent two vectors (x and y) which, when added together, give you a vector which goes from the origin to a point on the Cartesian plane. So although the destination is a point, the coordinates represent a vector. If you change your reference point (origin), the coordinates of that point would change (you would have a different vector) even though the point is the same.(48 votes)
- Would temperature represent a vector quantity? Let's say temperature rising from 10C to 20C?(32 votes)
- No, temperature is known as a scalar quantity. All scalar quantities give us a single value (eg speed, length/width/height, etc) that don't necessarily give us a direction. Vectors on the other hand are quantities which have direction as well as magnitude, esp. when trying to find the position of one point in space relative to another.(40 votes)
- When you place numbers on a number line, they have a magnitude and direction from the origin. As far as I can see, they are just the same as points on a cartesian plane, except in one dimension rather than 2. So why can't you say the number (+)5 is a vector?
For example, when I try to explain how negative numbers work, I draw them as arrows with the same magnitude as corresponding positive numbers, but pointing in the opposite direction. Then I can show that subtraction is just reversing the direction of the vector you are subtracting (i.e. negating it) and then adding the vectors. That helps me to explain visually why subtracting a negative number is the same as adding the positive number of the same magnitude.(20 votes)
- You're right in that +5 and -5 can be the same as denoting east or west on a number line in some cases. And it is reasonable to think of a signed number as a vector in 1-d space, yet only in certain circumstances. However, the very definition of a scalar is (1) a real number rather than a vector (2) a quantity (as mass or time) that has a magnitude describable by a real number and no direction. Real numbers can be negative or positive.
The reason that it's necessary to differentiate between vectors and scalars is two-fold (and perhaps there are more reasons as well) 1: Scalars obey algebraic rules for the operations like addition and multiplication. And vectors obey vector algebra for the operations like addition and multiplication. 2: A scalar can divide another scalar. And two vectors can never divide each other.
Also +5 in itself doesn't necessarily mean that it's on a number line that is meant to show direction. (ex. east-west) It could mean anything from how many apples you ate to the score of your team in a ball game. If I ever had to consider +5 as a vector I would only do so if wanted to know the displacement from zero (or any other number) on a number line. Temperature is a common example of scalars and it has negatives and positives and interestingly is shown on a number-line-like scale.(10 votes)
- How the point (5;5) can represent a vector when it is a quantity in a two-dimensional space? In what way does it show a direction?
I mean when you throw a ball in a three dimensional space it does have a magnitude and a direction. But how a point on a two-dimensional coordinate plane can have a direction? To me, how I see it, it is just an end point on a coordinate plane. If a point would show an arrow with a direction, then I would see it as a vector. I do not really see how it is a vector?
Any help would be appreciated.(15 votes)
- You can use the point (5,5) because it gives a 45˚ angle from the point (0,0) in the positive X, and a magnitude of √50.
Note: This is almost the same explanation as my answer to Daniel Lozada. Please don't blame me for not being creative.(4 votes)
- Would a vector classify as a line, ray, line segment, or neither?(8 votes)
- No, Sal was using the x and y coordinates just to explain to you. A vector is essentially just a figure that represents a direction and a magnitude. A line, ray, or segment does not represent any of them. They're for a line or segment will not be a vector quantity. But if you said the magnitude of the line then that would be a scalar.(7 votes)
- In Sal's example, he talked about how "the angle measure 5 degrees" was not a vector because it only had a direction. But, couldn't the 5 degrees be the magnitude?(5 votes)
- Not in this context. Yes, 5 degrees is the magnitude of the "rotation" relative to an axis, but it doesn't say what is 5 degrees away or how much of it there is. Saying a point is 5 meters away only tells us that it lies somewhere on the circle 5 meters away. Similarly saying a point is 5 degrees from us only tells us the direction. It could be 2 meters away or 1000 meters away.(8 votes)
- Can a magnitude be negative or 0?(4 votes)
- No, a magnitude is always positive. You can think of it as finding the absolute value from one point to another-- no matter where the points exist, the distance between them will always be positive.(4 votes)
- How do you use vectors ?(5 votes)
- One way you can use vectors is when you calculate velocity. Velocity is speed with a direction. For example, if a car is moving 25 mph to the right, then you have the car's velocity, which is also a vector (since it has magnitude and direction. If you were only given the fact that the car is going 25 mph, then you have a scalar, which only has magnitude/size. See this video for a better understanding about vectors and scalars: https://www.khanacademy.org/science/physics/one-dimensional-motion/displacement-velocity-time/v/introduction-to-vectors-and-scalars. More examples (a quarterback's pass, wind, and billiard balls) are listed on http://www.scientificamerican.com/article/football-vectors/.(2 votes)
- If (5,5) can represent a vector by virtue of the cartesian coordinate system then 5+5 can represent one on the number line, no?(2 votes)
- Would temperature represent a vector?(2 votes)
- No, temperature is a scalar quantity. If an object is 40º C, it isn't 40º C to the left, or north, or down. It's just 40º C.(2 votes)
Voiceover:Which of the following can represent a vector? A vector is something that has both a magnitude and a direction. Let's see which one of these could represent something that has a magnitude and a direction. The first choice right over here is the number 5. 5 could represent a magnitude, you could say, essentially, how large something is. That's all the information has. It doesn't say 5 in a certain direction. This one by itself would not be a or it couldn't represent a vector. You need to specify a direction as well. The angle measure 5 degrees. Well the angle measure 5 degrees could represent a direction. If you say it's 5 degrees. Let's say that's the positive X axis that's the positive Y axis. If you said it's 5 degrees relative to the positive X axis. If you said something like this, so if you had 5 degrees over there, that would be specifying a direction, but it's not giving us a magnitude. It's not saying how far in that direction or how large in that direction. An angle measure of 5 degrees, you're giving us a direction, but no magnitude. Let's just cross these out. The point (5,5). Now this is interesting. If we say that the point is the end point of a vector that starts at the origin, so lets draw that out. That's the positive X axis, positive Y axis. This is 1,2,3,4,5 in the horizontal direction. 1,2,3,4,5 in the positive vertical direction. The .5,5 is going to be right over there. If you said that this represents the head of a vector who's tail is at the origin. The vector would look something like this. Now, you're giving both a magnitude and a direction. What's the magnitude? What's the distance between the origin and this point, right over here. You could solve using the Pythagorean theorem and the direction, it's this general direction. This could specify a magnitude and a direction. This could represent a vector. The outcome of 5+5, obviously that's just 10. That's just a number. For the same reason, this first choice, if I just have a number, that could be a magnitude, but we don't have a direction as well. This could not represent and that one could not represent a vector.