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# Adding vectors algebraically & graphically

To add the vectors (x₁,y₁) and (x₂,y₂), we add the corresponding components from each vector: (x₁+x₂,y₁+y₂). Here's a concrete example: the sum of (2,4) and (1,5) is (2+1,4+5), which is (3,9). There's also a nice graphical way to add vectors, and the two ways will always result in the same vector.​​. Created by Sal Khan.

## Want to join the conversation?

• Can we subtract vectors? •   Yes you can and it works in exactly the same way. That is subtracting the vector [3, -4] would be the same as adding the vector [-3, 4]. So subtracting a vector is the same as adding a vector that goes in the opposite direction with the same magnitude. It's a bit like when you first learn subtraction using a number line and see that subtracting a number mean moving left along the line whereas adding means moving right.
• So why the magnitude of a vector doesn't imply its origin; you say that we don't care about where the vector starts, but the magnitude, in order to be defined, doesn't need to have mentioned a point of departure and a point of arrival? • At , what is meant by the sentence that vector a and vector b belong to R2. What is meant by R2? Please i want an immediate response!! •  R2 represents all the real numbers in a 2-Dimensional world.
So any RN represents all the real numbers in a N-Dimensional world.

Hope this helps,
- Convenient Colleague

Sorry that I get here 5 years late...
• What is rectangular form!? • A vector in rectangular form is when you specify the components of the vector along each of the rectangular cartesian coordinate system axis, so you can specify a vector as a tuple of numbers: `(a, b)`, or using unit vectors along the axis: `ai + bj`.

Another way of representing vectors is in polar notation, where you give the length of the vector and it's angle measured form the positive x axis, that is usually called "polar form".
• Does the vector always start at the origin? • It doesn't have to start at the origin because it doesn't matter where it starts. A vector doesn't have a "starting point" or "ending point". It only has a magnitude and direction. A vector that is REPRESENTED as starting at the origin is the same vector that is REPRESENTED as starting anywhere else, as long as the magnitude and direction are the same. Note that vectors aren't really equivalent vectors, but the same vectors. They are just represented at different places.
• At and before Sal show different vectors and how you can draw them. Is this really necessary to know how to draw all these different vectors? on the "graph"or can I just draw the vector i am most comfortable with. • He is showing how you can draw the same vector on different coordinates (better to say the same vector scaled for a given scalar). It is not necessary to draw all those different vectors, but it is useful to know you can draw them that way. Also, when you solve problems which include the addition, substitution etc. of vectors you are always going to draw the vectors which have certain coordinates or given dots (A, B, C...). You can always "put" them on the position where you need them to be.
• Here we form a triangle by using head to tail rule. When do we use the parallelogram rule? • Exactly how can you use vectors in the real world? • Well, vectors are basically everywhere! When we kick a ball in a projectile motion, the velocity of the ball is a vector (it rhymes). Programmers also use vectors in their programs so that the animation of objects look real. NASA uses 3D vectors to plot courses for their space exploration robots.
• how do you find the magnitude of a vector? • You can think of it as finding the hypotenuse in a right triangle. For example, we can have a vector "v" that begins at the origin and terminates at point (-5, 12). We can create a right triangle in which the vector is the hypotenuse, so we can use the Pythagorean Theorem.

c^2 = a^2 + b^2
c = sqrt(a^2 + b^2)

|v| = sqrt(x^2 + y^2)
|v| = sqrt((-5)^2 + 12^2)
|v| = sqrt(25 + 144)
|v| = sqrt(169)
|v| = 13 