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## Precalculus

### Course: Precalculus > Unit 6

Lesson 5: Vector addition and subtraction# Subtracting vectors end-to-end

Watch Sal determine which diagram represents vector subtraction graphically. Created by Sal Khan.

## Want to join the conversation?

- What are vectors commonly used for in real life? Why are they useful?(39 votes)
- Vectors are used heavily in Physics, where many of the fundamental quantities (displacement, velocity, acceleration, force, momentum, impulse) are directional. The vector incorporates both the magnitude and direction of the quantity.(55 votes)

- Aren't those triangles the same? Would it matter which direction it's going? Shouldn't you just lay the up flat on a line? Sorry for asking so many questions.(7 votes)
- Yes, the direction of the arrow is important. That is the difference between driving 10 miles East and driving 10 miles West. They might be the same distance, but they take you very different places.(30 votes)

- How you add subtract vectors?(8 votes)
- You don't add subtract vectors, but add to a negative vector.(6 votes)

- How is this used in real life?(2 votes)
- computer graphics, physics, abstraction. Mathematics is training for a problem you haven't come across yet.(13 votes)

- How do you know which direction the answer vector is going?(3 votes)
- For vector addition, you would translate the second vector such that the "start" point of the second vector coincides with the "ending" point of the first vector. For subtraction, you would first flip the second vector (or whichever vector is being subtracted) such that the arrow head is on the other endpoint of the vector, making it point the opposite direction. Because you have just negated your vector (flipped its direction), you are essentially now subtracting a negative, or adding the flipped vector. When you perform the vector addition using the flipped vector, you will figure out the direction--the direction of the resultant vector is from the start point of the first vector to the ending point of the second vector (when translated correctly).(5 votes)

- It's not the video itself that's the problem. It's that the preceding videos don't set it up, motivate it, or explain it. This was true of others in the vector series. It's not nearly as week designed as the sequences on other topics.(4 votes)
- I agree with you! This section of videos was more "learn by an example" with out much in the way of explanation.

As far as vector subtraction is concerned, there is just a simple rule to remember, which this video tried to show, but was not explicitly clear about how to set up the subtraction nor the difference between u-v and v-u.

Suppose you have vector v and vector u. The head of the vector is where it starts, the tail of the vector is where it ends, that is, the arrow-head end.

To draw v-u, align the head of v to the tail of u and the result vector is drawn from the head of u to the tail of v.

To draw u-v, do the opposite: align the head of u to the tail of v and the result vector is drawn from the head of v to the tail of u.(0 votes)

- How do I know if I have to "flip" a vector in order to write an equation?(2 votes)
- To draw the negative version of any vector, we keep the magnitude constant, but just change its direction to the opposite, which is flipping.(3 votes)

- What are rectangular vectors?(2 votes)
- Vectors being measured by the left-right and up-down dimensions , versus polar ones that are determined by a radius (distance from starting point) and an angle (from a pre-determined line).

When you follow compass you use polar vector by moving "this" many steps forward in a direction that is such-and-such an angle from North. When you use a map that says go 5 blocks along this road and 3 blocks on that road, you are using rectangular vectors.(2 votes)

- In the next practice, how are we supposed to write vector V for example using our keyboard ?(2 votes)
- You don't have to add any special symbol. Just type in v and it will accept it.(2 votes)

- how do i write a vector symbol on my keyboard?(2 votes)
- If you're talking about the pointy brackets <>, they're above the comma and period keys. Most standard keyboards don't have an easy way to put the half-arrow over a letter, if that's what you mean. As with the symbols for a ray, line segment, etc., you'd need special fonts or programs like LaTeX.(1 vote)

## Video transcript

Voiceover:These are
vectors A and B, all right? That's vector A, that's vector B. Which of the following describes a valid way of obtaining vector A minus vector B? So let's look at our
choices right over here. Here they just depicted
vector B, just like it's depicted here, it's maybe
shifted over a little bit. And then over here,
they depicted vector A, the negative of vector
A, and it looks like they're taking the negative of vector A, and to that they're adding
vector B, and then they're claiming that this magenta
vector is A minus B. If we ignore what they
wrote over here, what they've done is they've
added negative A to B. So this is the negative
of vector A plus vector B. Or we could think of it as
vector B minus vector A. So this one right over here is not right. If we swap the signs,
if this was a negative, if we put negative A plus
B, then we'd say okay, this is accurate, but
they want us to figure out valid ways of obtaining A
minus B and this isn't that. This is B minus A. Let's look at the other choice. You have vector A. Let's start at the tail of vector A, get to the head of vector A. Over here we have the tail
of the negative of vector B, so they essentially flipped
it over when you take the negative of it, and so we're
adding negative B to A. If we take the tail of
where we started to the head of where we ended, yes,
this would be vector A plus negative vector B,
which is the same thing as vector A minus vector
B, so that one is right. That's a valid way of obtaining
vector A minus vector B.