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### 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?
• 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.
• 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.
• 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.
• How you add subtract vectors?
• You don't add subtract vectors, but add to a negative vector.
• How is this used in real life?
• computer graphics, physics, abstraction. Mathematics is training for a problem you haven't come across yet.
• How do you know which direction the answer vector is going?
• 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).
• 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.
• 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.
• How do I know if I have to "flip" a vector in order to write an equation?
• To draw the negative version of any vector, we keep the magnitude constant, but just change its direction to the opposite, which is flipping.
• What are rectangular vectors?
• 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.