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## Precalculus (2018 edition)

# Equivalent vectors

Sal determines if two vectors shown on a graph are equivalent by seeing if they have the same magnitude and direction.

## Want to join the conversation?

- What does it mean to put a vector inside 2 bars on each side?

(llwll)(17 votes)- Yes, it means the magnitude or the size of the vector. For example, if you were going in a car and you apply brakes as you see a stop signal, the car will decelerate at a rate. That rate can be -5 meters per second square or anything. This value (-5) is the vector quantity and the magnitude of it would be 5, since the absolute value of -5 is 5. If you're unfamiliar with absolute values, check out khan academy videos on them.

Hope it helped :)(3 votes)

- If you eyeball it, the vectors do seem parallel. But is there a sure-shot method to check if the vectors are parallel? If yes, please specify.(8 votes)
- You can determine if two lines are parallel by finding the slope of both lines. If you find that the slope of each line is equal, then you MAY be parallel. If the slopes are exactly the same, the two lines could either be parallel or right on top of each other. You have to find the y intercept (set x=0 when you have the line in standard form y=ax+b and solve for y) of the two lines. If the y intercepts are different, then the two lines are parallel. This is because the lines will be rising and running (have equal slope) at the same rate, but aren't on top of each other.(17 votes)

- Isn't the slope the magnitude of the vector?(4 votes)
- No. The magnitude of these vectors is found the same way we find the length of the hypotenuse of a right triangle using the Pythagorean theorem. Pretend the vector is the hypotenuse of a right triangle. Break the vector down into its horizontal and vertical components as though they were the legs of that right triangle, square them, add the squares, and take the square root of that sum. That is the magnitude.

The magnitude of the vector is the square root of the sum of the squares of the vertical and horizontal components. The slope is really the direction.(12 votes)

- But how does one say that they are of the same magnitude??

just by the nature the it goes top right?? that says they are both in the sane direction??(5 votes)- Magnitude is the length of the vector, which Sal finds out in the video using Pythagoras' Theorem. This is also the basis for the
*Triangle Law of Vector addition*. The direction of 2 vectors is the same, if they act *along the same line, OR along 2 parallel lines.*

Assuming A and B to be vectors,`A = B iff | A | = | B | & direction of A || direction of B.`

(Since 2 lines on top of each other are parallel to each other.)(7 votes)

- Should I Master mathematics in order to study physics in more depth?(5 votes)
- Yes. I am taking physics and calculus at the moment and am amazed at how much that is taken for granted in physics can be explained and can be derived through calculus or other forms of math. Also, I believe that physics is just applied math, and to learn how to apply math, you will need to know how it works.(5 votes)

- How exactly can vectors be equivalent with just the same magnitude and direction (Like at around5:50)? Why don't the location of the vectors (their initial and terminal points) affect whether or not they are equivalent?(4 votes)
- Because a vector is defined by its Magnitude and Direction, but the starting point can be any where, unless it's specified to be the origin, hope that help .

sorry fort late reply(2 votes)

- if two vectors while having same magnitude and direction is that equivalent(3 votes)
- Well not really. Only vectors with the same components can be computed meaning a vector with an X-component can be added with another vector with an X-component also meaning u can't compute vectors with different components as in adding or subtracting an X-component with a Y-component. It is after u find the computation result of X and Y component that u can find the resultant. So unless the components are the same two vectors with unknown components can't be equivalent.(1 vote)

- What is the difference between "IIuII" and "IuI" ?? How are they equal??(2 votes)
- These are two types of notation to mean the same thing in vectors. More often you will see ||u|| in vectors for the magnitude, and |u| for a matrix u to talk about it's determinant. Since they are very much related, the notation occasionally crosses over from one to the other.

The magnitude is equal since the they have the same length. (use the distance formula, or Pythagorean theorem).(3 votes)

- At2:17, what does the triangle beside y indicate? Is this symbol used for other things too?(1 vote)
- You're right, this symbol is in fact a very important symbol in most of mathematics. It is the Greek letter delta (mostly Δ, sometimes 𝛿) which, at least in this case, means 'change in'. Think of it as just an important fancy symbol because as you go further into calculus, this symbol will keep showing up.

In the video, Sal writes 'Δy' (read as delta y) instead of 'change in y' because it is the more formal notation. (just like the arrow above vectors)(3 votes)

- Would two Vectors still have the same magnitude if one is (5,3) and the other is (3,5) or (-3,-5)?(2 votes)
- Yes, that is correct. These examples of vectors do hold the same magnitude, but they will have different directions.(2 votes)

## Video transcript

- [Voiceover] I have some
example problems here from our equivalent vectors
exercise on Khan Academy, so let's go through these
and like always, pause the video and see if you can
work through them on your own. So this first one says, "Are
vectors u and w equivalent?" And so we can see vector u here in blue and vector w right over here. And we have to remember a
vector is defined by both having a magnitude and a direction. And so for two vectors to be
equivalent they have to have the same magnitude and the same direction. So when I look at these
two, they are clearly pointing in different directions. Vector u is pointing to the bottom right, that's the direction it's pointing in, and vector w is pointing
to the bottom left, so they definitely aren't equivalent. So they are not equivalent,
scratch that out. So they have different
directions, different directions. Different directions. And also think about the magnitude. If I just look at the
length of the arrows, just eyeballing it,
they look pretty close. Let me verify that. So if I'm starting at the
initial point for vector u, how much do I move in the x direction? Well in the x direction
I go from negative eight to negative three, so I
could say my change in x is positive five, my x increases by five as I go from the initial point,
from the x coordinate of the initial point to the x coordinate of the terminal point, so that length. The magnitude of just
the x component is five. And let's see what happens
in the y direction. So in the y direction, I start
at y equals negative two, right over here, and then I go down to y equals negative eight. So my change in y, change in y, is equal to negative six, negative six. Also think about this one over here. What's my change in x? What's my change in x? Well I'm starting at x equals eight, and I am going to x equals three. So my change in x is negative five. I'm gonna write that,
change in x is equal to, this is my change in x, change in x. And then what's my change in y? Well I start at y is equal to eight and I go down y is equal to two. Do it just like that. So my change in y is
equal to negative six. Now based on the changes in x's and y's, I can figure out the magnitude
of each of these vectors. The magnitude of u, so I
could write it like this, the magnitude of u, sometimes you'll
see the notation like this, the magnitude of u, of
vector u I should say, is going to be equal to, we're just gonna use the pythagorean theorem here. It's going to be the square root. The length of this hour is just a hypotenuse of the right triangle. So it's going to be the
square root of five squared plus negative six squared so
it's gonna be the square root of 25 plus 36, which is
equal to the square root of, what is that? It's going to be square root of 61. I can make that radical
a little bit smaller, square root of 61. Now what about vector w? So the magnitude of vector w is, I could use a double, those double bars. The magnitude of vector
w, well that's going to be the square root of negative
five squared, which is 25 plus negative six squared,
which is going to be 36. Well that's going to be the
square root of 61 as well. So they have the same magnitude,
just different directions. So let's see, no they have the same magnitude, but different directions. Yup, that's the choice we like. Let's do one more of these. Are vectors u and w equivalent? Alright, so these look pretty
equivalent just eyeballing it. It looks like they're pointed
in the same direction, they're going from the
bottom left to the top right. And it looks like they're the same length, but we can verify them. Once again by looking at
their x and y components. So in the x direction, or
thinking about how much we change in x and change in y when we go from the initial point to the terminal point. So in the x direction, for
vector u, we go from negative seven to negative four,
so we increased by three. And what do we do over here? We go from x equals two to x equals five. So our x component once again
has a magnitude of three, our change in x is a positive three. And so our change in y,
what is that going to be? Well we're going, on this
vector, from y equals... Whoops, I'm using the wrong... We're going from y is equal
to one to y is equal to six. So the magnitude of the y
component, we could say, is five or our change in y is a positive five. And then the change in y over
here also is a positive five. We go from negative seven to negative two. It's also a positive five. So notice our change in x is the same, it's a positive three. Our change in y is the
same, positive five. So that let's us know that we have the same magnitude and direction. And so these are equivalent,
these are equivalent vectors. And, in fact, as we'll see in
the future, you can actually denote a vector by its components. We could say that vector u,
vecotr u, is the vector five, oh sorry, the vector three comma five, our change in x comma our change in y. And that is exactly the same as vector w. So that is exactly the same as
vector w, has the same change in x, same change of y as
you go from the initial point to the tip of the
arrow, the terminal point.