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Interpreting statements about vectors

Vector quantities have both a magnitude and a direction. In this example, we interpret a mathematical statement about two vector quantities in terms of the real-world quantities they represent. Created by Sal Khan.

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  • piceratops ultimate style avatar for user Denis Orlov
    Does this ||a||=||b|| not mean its the same velocity? How is it different from a=b. Shouldn't the answer be A?
    (2 votes)
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    • aqualine ultimate style avatar for user Simum
      ||a||=||b|| means the magnitude of a = the magnitude of b. But a=b means the magnitude and the direction of a = the magnitude and the direction of b. And so, the answer is B as their magnitudes are the same but not necessarily their direction. Hope that helps.
      (5 votes)
  • starky seedling style avatar for user deka
    then how can we represent the option C in a mathematical manner?

    before checking the answer, i assumed |vector A| = |vector B| might say same magnitude different directions (same as in the case of scalars), while ||vector A|| = ||vector B|| saying different magnitudes same direction

    i googled it real quick but found nothing yet
    please help

    anyhow thanks for your service, Sal as always
    (1 vote)
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    • leaf green style avatar for user kubleeka
      The "proper" way to denote the magnitude of a vector A is ||A||. We sometimes write |A| as shorthand. So if A is a vector, then |A| and ||A|| will always denote the same quantity.

      Two vectors A and B point in the same direction if one is a scalar multiple of the other, if there exists a scalar k such that A=k B.
      (1 vote)

Video transcript

- [Instructor] We're told that particles A and B are moving along a plane. Their velocities are represented by the vectors, vector a and vector b respectively. Which option best describes the meaning of the following statement. Choose one answer. So pause this video and try to work through this on your own before we work through this together. All right, now let's work through this together. So this is saying that the magnitude of vector a is equal to the magnitude of vector b. So we know that a vector is specified by both a magnitude and a direction. And this is just saying that the magnitudes are the same. So for example, vector a could look like this, and vector b could look like this. It could do something like that where it has the same magnitude and the same direction. Or vector b might be in a completely different direction. The magnitudes being equivalent just tells us that the length of these arrows are the same, but we could have different directions here. So what this tells me is that we have the same speed which is the magnitude of velocity, but not necessarily the same direction. Now let's look at the choices here. The first choice is that two particles move at the same speed and in the same direction. So we've already said that that's not necessarily the case. In order for choice A to be correct, they would essentially have to be equivalent vectors. Choice A would be the case if we said that vector a is equal to vector b, then they would have to have the same magnitude and the same direction, the same magnitude and the same direction. But that's not what they told us. They just told us that the magnitudes are the same, not necessarily the directions. So I'll rule that one out. The two particles move at the same speed, but not necessarily in the same direction. Yes, that's what we just talked about. They have the same speed, which is the magnitude of velocity, but they didn't tell us anything about the direction, just the magnitudes. So I like this choice, but let's look at choice C. The two particles move in the same direction, but not necessarily at the same speed. Well, we know they move at the same speed. That's what this is telling us. The magnitudes are the same. We just don't know anything about the direction. So I would rule this one out as well. In order for choice C to be the case, you would see something like this, maybe that is vector a right here, and then vector b would move in the same direction, but it would have a different magnitude. And here you would visualize the magnitude as the length of the arrow. But that's not what they told us. They told us this right over there.