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# Vector triangle inequality

Proving the triangle inequality for vectors in Rn. Created by Sal Khan.

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• why should he stop saying magnitude. When the definition of a vector is having both magnitude and direction. and the positive or negative indicates direction and the magnitude is the the absolute value of the vector^2 which really is neither positive nor negative.? • If I understand Sal, he is trying to avoid the confusion between scalar and vector quantities. A vector has both length and direction - what is referred to as magnitude. (The vector [x] (1, 2) has exactly the same length as vector [y] (-1, -2), but a different direction, hence a different magnitude, and therefore, [x] <> [y]). On the other hand, the length of the two vectors is equal, hence ||x|| = ||y||. In the process of calculating ||x||, all information regarding direction is lost.

A good analogy can be found in physics in the distinction between speed and velocity. An object in uniform circular motion has constant speed but has an ever changing velocity - because velocity includes direction.
• what about ||vector x||-||vector y||<=||vector(x+y)|| ? what happens when c is negative? can then it this part be proved? • Note that `||x|| - ||y|| <= ||x+y||` is a much less restrictive statement than `||x|| + ||y|| <= ||x + y||`. All of the lengths are (by definition) positive values. Multiplying one of them by -1 on the left side of the inequality just makes that side even less than before.

Perhaps it's helpful to think of the axiomatic statement: `-||x|| <= ||x||` since all "negative or zero" values are less than or equal to all "positive or zero" values.
• Does the equality work if vector x or y are zero? I know we assumed they were Not Zero for the proof, but it seems like if we let either one or both be zero, it still holds true. • At , Sal says "I should stop using the term magnitude." Why? • Why is x dot y <= | x dot y | ? • Sal corrects himself a couple times when he says "magnitude", saying that he should say "length" instead. But I thought magnitude and length are the same thing? • what is the difference between the length and the magnitude of a vector? • When adding two vectors in n-space, don't the three points that define this addition (the tail of the first vector, the head of the first vector/tail of the second vector, and the head of the second vector) just define another plane (or set of planes along the same line for the degenerative case) that's in some equivalent version of R2? So it seems to me that these relationships between pairs of added vectors are always in an R2(-like) space. So I'm not really sure what Sal means about extending these definitions beyond R2. • Dave, you are correct in your observation that the three vectors considered are all co-planar. This observation is very helpful for having an intuitive grasp of what is going on.

Mathematically, however, an R2-like space is not a well-defined concept. Consider our n-dimensional vectors as before. They are co-planar. We could perform a change of basis, such that the vectors lie along the x-y plane, and then it would act much like in the 2D case, so we can use two dimensional reasoning. However, by extending to Rn, this essentially picks up the flat plane of these vectors and puts it at some angle in n-dimensional space. It is not necessarily immediately obvious that the situation can be reduced to two dimensions.

Extending to n-dimensions is more useful if you have more vectors. Say we have more than the vectors x, y and x + y. Say we have another vector which is not co-planar to these vectors in our problem, and we wanted to do some maths or operation which included the triangle inequality as one of the steps. If the triangle inequality only applied to two dimensions, then we would not be able to use it in this case. However, since we have proven it in n dimensions, we can use it even if we have other vectors in the problem which aren't co-planar.  