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Visualizing vectors in 2 dimensions

Visualizing, adding and breaking down vectors in 2 dimensions. Created by Sal Khan.

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• What are the strange ‖ symbols that keep popping up? They look like 2 small vertical lines together.
• That means the magnitude of a vector. denoted ║a║, were a is a vector.
• Why is it so hard to imagine the fourth dimension?
• Another thing is, we can only see our dimensions, and those are the 3. We can not imagine 2 dimensions either, because say it was height and width, you could not see it in out dimension, it would not have depth, making it invisible to our eyes. It is the pretty much the same think with the other ones.
• I got confused for a bit thinking he put a load of elevens everywhere but then I realized they where just lines to make it a bit neater lol. Or where they for something else? E.g where it said II a II=5. I haven't done any trigonometry yet either.
• ||a|| is just magnitude.
That means you can forget the direction.
ex. - acceleration due to gravity is -10m/s^2 because it is in downward direction.
but the MAGNITUDE is 10m/s^2.
and thats the same thing as ||a||.
It is also sometimes written as |a|
• why are the variables put between || || ? like ||a|| for example.
• || a || represents the scalar component of a vector. Remember that a vector has magnitude AND direction, while scalar quantities ONLY consist of magnitude.
EX: acceleration (a)= 30m/s/s to the RIGHT is a vector; || a ||= 30m/s/s is scalar
• Is it possible to have a vector in 4 dimensions? If so, how would it look?
• If one accepts that time is the 4th coordinate (the 4th dimension), then it is necessarily a piece of the context of vector. As far as what it would "look like", that's a little trickier (as if that first statement wasn't ambiguous enough..). Try to stick with me on this though. Time is a way of comparing the change of other objects to some constant(s). On Earth, we use our motion around the sun as our constant. Further, we use metrics like "meters", "grams", etc, as constants. When you are observing a given space (picture a model of planetary orbit around the sun or a shoe-box diorama for that matter), it will "look" however it "looks" when your potential coordinates are all satisfied in relation to the constants. For example, in the year 2025 (2,025 revolutions of Earth around the sun after the life/death of "J.C."), Earth will be at spatial coordinates x,y,z. Once you are at this particular coordinate though (x,y,z,2025), you can only speak of what the vector was that got it there, and what it will be (assuming "ceteris paribus")
• I still don't understand how A + B = C!!
• Try taking the vectors apart and looking at their components. If it's like this, you often can visualize the addition better. When we put vectors from tip to tail in order to add them, it's like we're separately adding the vertical components and horizontal components, and then condensing that into a new vector.
Say we have a vector pointing straight up, and another vector pointing up and rightwards (excluding the specific information and magnitude to make the problem clear). To add them graphically, you would take the straight up vector and put the tail of the up-and-right vector onto the tip of the up vector. Let's now do this with their components. The vertical component of the up vector is added to the vertical component of the up-and-right vector, creating a new vertical component that's even greater. The horizontal component of the up vector is 0, so the new one would be the same length as the horizontal component of the up-and-right vector. We then create the resultant vector and it is greater in magnitude than either of the two were, and its angle is in between that of the up-and-right vector and the up vector.
• Is the 4 dimension time?
• Yes, sort of. There are three spacial demensions and one time demension. None is exactly the first, second, etc. It's like, if you have 4 cups of water, which is fourth? Answer: none. they are all first.
• At , why didn't Sal just draw a line connect Vector A and Vector B, and why he needed to move Vector B to the head of Vector A?
• As he said in the video he was showing that a vector is a defined by a magnitude/length and a direction but the position of the vector in the coordinate system is irrelevant to the definition of the vector.

He probably started out with the vectors starting at the same point because you often have diagrams like that where you are showing the forces on an object, a good example is a free body diagram.

He moved the tail of one vector to the head of the other because that is the geometric way of looking at what it means to add vectors.