Algebra (all content)
Worked example: Scaling unit vectors
Watch Sal scale up a unit vector to have a magnitude greater than 1. Created by Sal Khan.
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- How do you scale up a vector?(8 votes)
- You multiply your equation by a number, like in the video.
For example, if vector U coordinates are (3, 7) and if you want to scale it by two, then you multiply the coordinates by two. Like, 3*2 and 7*2, then your new coordinates of your vector would be, (6, 14).(24 votes)
- previously Sal explained two unit vectors- i^ and j^. How are these unit vectors different from them??(6 votes)
- The unit vectors i and j usually refer to the specific unit vectors <1, 0> and <0, 1>, which are oriented horizontally and vertically and each have magnitude 1.
The unit vectors that we find here could have different directions (since the problems often ask us to find a unit vector in the same direction as another vector), but they have to have magnitudes of 1. The main difference between these and i or j is just the direction that they point in.(4 votes)
- Is it possible to scale with negative numbers?(6 votes)
- Yes, a negative number reverses the direction and scales; so -1 would just reverse the direction, -5 would reverse and increase the vector 5 times in magnitude, etc. . .(10 votes)
- What is the need to introduce this concept of unit vector?(5 votes)
- Suppose, a question asks us to find a vector with some magnitude, say 3, and the direction same as that of a vector, say 2i+3j+3k. Now, since given vector has a direction, which we have to consider, along with a magnitude, which we do not have to consider. Thus to get rid off that magnitude, so that we could simply multiply the given magnitude with the unit vector of the given vector, we use unit vectors.(5 votes)
- at 0.17 Mr Sal Khan used pythagoras theorem to find the magnitude of the vector but in some previous videos he added both horizontal and vertical components. Should we use pythagoras theorem or add both the vectors. I am confused .(1 vote)
- Vectors have both a magnitude and a direction. Adding hor and ver will give you the direction of the resulting vector, but the Pythagorean thm will give you the magnitude of the result.(5 votes)
- can we multiply vectors by vectors ?(3 votes)
- Is it possible to scale up the 'x' and 'y' components by different factors?(2 votes)
- This would end up changing the direction of the vector. Usually, when we scale a vector, we wish to preserve its direction and only change its magnitude. Otherwise, there's really no significant relationship between the original vector and the "scaled" one.(1 vote)
- Is it a convention to write i first instead of j?(1 vote)
- No, it's not a convention. Unit vector I is defined as a shift of 1 in the horizontal direction (1 to the right). Unit vector j is defined as a shift of 1 in the vertical direction (1 up). Since vectors are defined as ( shift in x, shift in y) the i must come first.(3 votes)
- Why do we learn this? I mean when do we use this in real life other than when you are a math teacher?(1 vote)
- Can we divide vectors even though we can not divide matrices or will we be multiplying one vector by the inverse of another?(1 vote)
Voiceover:Let's say that I have the unit vector U and in the horizontal direction for every 1/3rd it goes, it goes square root of eight over three in the vertical direction. And we can verify that this is indeed a unit vector. The magnitude of our vector U is going to be equal to the square root of the sum of the squares of the components. And this, once again, just comes straight out of the Pythagorean Theorem. So it's going to be equal to the square root of 1/3rd squared plus square root of eight over three squared. And what's that going to be? Well that's going to be equal to the square root of one over nine plus eight over nine. Which is equal to, and I think you see this coming, square root of nine over nine, which is equal to one. So this is indeed a unit vector. Let's say someone says, "Well I like this direction, "but I don't want the magnitude to just be one. "I want to find some vector V "that has the same direction, so it has same "direction as U, but has a magnitude of eleven." So we want the vector V to have a magnitude of eleven. So how could I define vector V? One way to think about it is if we just scale each of the components of U up by eleven, then we will have gone in the same direction, but now we will have a magnitude eleven times as big. And if we start with the magnitude of one we'll now have a magnitude of eleven. So we could say that vector V could be eleven times one third, comma, square root of eight over three. And so what is this going to be equal to? This is going to be equal to eleven over three, comma, eleven square roots of eight, over three. And I encourage you, if you don't believe me, I guess on one level it should make sense that when you multiply a vector times a [scalerite] like this, it just scales it in that direction by this factor, so if your magnitude was one, your magnitude will now be eleven in that direction. But if you want, you could actually verify mathematically that the magnitude here, if you were to calculate it, is going to be eleven instead of one, as in the case of unit vector U.