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## Get ready for AP® Calculus

### Unit 8: Lesson 5

Scalar multiplication# Scalar multiplication: magnitude and direction

Sal analyzes the magnitude and direction of vectors that are a result of scalar multiplication of a vector whose magnitude is given.

## Want to join the conversation?

- At1:46Sal said that the magnitude can only be positive. But shouldn't that be non-negative?(36 votes)
- I'm sorry, I don't understand why that matters... what's the difference between positive and non-negative?(5 votes)

- what if the x and y have different multiples? Like (5x,8y)?(12 votes)
- No ones replied to you so I thought i'd give it a go...

Basically, you're not multiplying the vector by a scalar. You're multiplying its direction components by two separate scalars, if you want to think of it like that. It's the same principle however, you would just do √(5x)²+(8y)² ... we can't really imagine what this would look like however, as we don't know what the initial x,y components of the vector looks like. Although if you'd like an example:

Say vector 'V' has the direction components 3i + 4j then the magnitude of the vector would be √(3)²+(4)² which would equal 5. When applying your multiples (5 and 8) to each of the x, y components it would become √(15)²+(32)² which gives you 35.34 roughly...

So as you can see when multiplying the components by separate variables, it's no longer a simple scalar multiplication of the vector where we can just multiply the initial magnitude by say 3. Hope this clears things up.(12 votes)

- If a vector can have zero length, what would be its direction?(8 votes)
- Vector having zero length means it hasn't moved at all. Therefore, giving such vectors direction would be "pointless".(8 votes)

- how would you solve it if lets say that vector z was (-2x,3y)?(4 votes)
- First of all you would only have -2x and 3y if you were trying to write a Cartesian equation from the vector. In fact the vector z that you have specified is actually (-2,3) not (-2x,3y).(4 votes)

- For the second part of the first question, shouldn't the vector of z equal -10 not 10?(2 votes)
- No, because magnitude can't be negative.

Here some answers about negative magnitude https://www.khanacademy.org/math/precalculus/vectors-precalc/vector-basic/v/vector-representations-example?qa_expand_key=kaencrypted_dd955e583fd4db5ac769fdd39dfc0f36_3be9a70a9550935b6542ee89c9c48c926e47285bde88f4fe064d56f24e405ea8fd3cead4b477f6e921925d5e81a2a5c8e8a8f43a28a7936a108ef62858424f5dc9c0dc3468125d9d26d815ffa12b5d754d12777366a5b1d56465454a2145b53d05ac5fae3fdbf35508ceb8ab5c328de4e62b4951718efdec6d2deaf1ac6168d555c310762792999a9bc7c3c074d283e1633cc35a29139aa595c932f938e434c7(4 votes)

- why does the vector have to be always with a letter and not a number?(2 votes)
- If we denoted vectors with numbers

the chance of mistaking the "vector number" for its magnitude will be pretty high, causing mistakes in the calculation(3 votes)

- magnitude of a vector is √x^2 + y^2

So, √x^2 + y^2 = 5

x^2 + y^2 = 25

3(x^2 + y^2) = 75

3x^2 +3y^2 = 75

√3x^2 +3y^2 = √75

ans = 5√3 = 8.66

What am i doing wrong here?(1 vote)- If you scale a vector by 3, <x, y> becomes <3x, 3y>. x is replaced by 3x, y is replaced by 3y.

So if the magnitude is √(x²+y²)=5, we replace x by 3x and y by 3y to get

√((3x)²+(3y)²)

=√(9x²+9y²)

=√(9(x²+y²))

=√9√(x²+y²)

=3√(x²+y²)

=3·5=15.(4 votes)

- At the very beginning, since the magnitude of vector v is equal to 5, could you say that (x,y) would be (5x, 5y)? Thanks.(1 vote)
- No, because the magnitude of a vector is equal to the distance from the beginning point to the end point. The components x and y are the individual contributions in the x direction and y direction that would push the beginning point location to the end location. So the components x and y would be the sides of a right triangle and the magnitude of 5 is the length of the arrow (vector) that forms the hypotenuse. There are pretty much an infinite number of ways to get 5 as the magnitude. We can use trig to help us, but in your case, we can use x² + y² = v² from Pythagoras to figure out the components.

However, after we have said that the magnitude is 5 if the components are x and y, we cannot multiply the components by 5 and have the same vector. That would be a new, bigger vector.(3 votes)

- If a vectors x and y components are multiplied by different scaler numbers then, how to find the magnitude?(2 votes)
- Zygoat, you have done the same. He was asking that if suppose earlier |a| was (3,4) what to do if it now becomes (6,9) say ?(1 vote)

- While we write the magnitude of a vector, why we use the double bar (//v//)? Could not we use the single bar(/v/)?(1 vote)
- Using single bars means the absolute value of something. We can use double bars to represent the magnitude of a vector. So the common notation for the magnitude of a vector is ∥v∥(2 votes)

## Video transcript

- [Voiceover] All right,
so we're told that vector v is equal to x comma y. They're not giving us the actual numbers for the x component and the y component. They're just saying that the x component is going to be the variable x. And the y component is
going to be the variable y. And the magnitude of vector v
is going to be equal to five. All right, I think I
can digest all of that. And it says, fill in
the blanks to complete the following sentences. So they're saying vector
w right over here, they didn't have to actually
write the word vector. They have an arrow on top of the w, so we know it's a vector. So vector w its components
are three x and three y. So its x component is
three times the x component of vector v. And its y component is
three times the y component of vector v. Instead of having a y, it has a three y. Instead of having an x, it has a three x. So w is essentially
vector v being multiplied by the scalar three. So its magnitude is
going to be three times the magnitude of vector v. So if vector v has a magnitude of five, vector w's going to
have a magnitude of 15. It's going to have three
times the magnitude. All right, now they tell
us vector z is equal to, so its x component is negative two times the x component of vector v. And its y component is negative
two times the y component of vector v. So you could view vector z as
negative two times vector v. Or the scalar negative
two times the vector v. And so you might say okay
that means its magnitude is going to be negative two
times the magnitude of v. But you have to remember, magnitude, that's always going to be positive. One way to think, when we talk about even absolute value we're really thinking about the magnitude of something. We're talking about how long it is. We're not thinking about its direction. So we wouldn't think
about the negative there. If we think about its direction, this negative is going to flip
its direction and we could think about that in a second. But if we just think about magnitude, we care about the two. So its going to have
two times the magnitude of vector v. So instead of five, it's
going to be five times two, which is ten. So now they say, match
each vector to defined to a vector that could be its equivalent. So if we think about it, the vector with the smallest magnitude
is going to be vector v. The other two we're
multiplying vector v by scalars with an absolute value of more than one. So they're going to
have larger magnitudes. So the smallest magnitude
is this one right over here. This is the shortest vector. So that would be our vector v. Now, vector w would be
one that has three times the magnitude and it's
going in the same direction. So, this one has three
times the magnitude. It looks like three times, roughly. And it's definitely going
in the same direction. And then vector z would have
two times the magnitude. But since we have, we're multiplying it by the negative scalar, it's going to go in
the opposite direction. And this looks like it. This purple arrow looks like
it's about twice the length of this red arrow. And it's going in the opposite direction. So I feel pretty good about
what we have done here. Let's do one more of these. So, let's do this one here. Here we have a vector v again. It has a magnitude of 10. And it says, fill in
the blanks to complete the following sentences. Vector w now, so now
vector w, it's taking each of the components of vector
v and multiplying them by one-fifth. So you can view vector w as
being equal to one-fifth, the scalar one-fifth times vector v. The magnitude of w is
going to be one-fifth the magnitude of v. One-fifth of 10 is just going to be two. Now vector z, it looks
like it's three-fifths times vector v. We're multiplying each of the
components times three-fifths. So that's like multiplying
the entire vector by the scalar three-fifths. So it's magnitude is
going to be three-fifths the magnitude of vector v. And so three-fifths of 10 is six. All right, then they say,
match each vector defined above to a vector that could be its equivalent. So here the vector with
the largest magnitude is vector v. This would be vector v. And the vector with the smallest magnitude is vector w. It's one-fifth the magnitude. So this has a magnitude, should be 10. This should one one-fifth of that which is a magnitude of
two and it looks like that. And this should be three-fifths of the 10, which is a magnitude of six. And that looks about right, so I feel good with
what we have filled out. So hopefully that gives you
a sense of how to tackle these types of example problems.