I know this should be very simple to understand but it makes no sense to me.
Why use for example 5i + 6j + 7k ?
And what is a unit vector compared to a normal vector? Online it says they’re used to “specify the direction of a vector” but I just don’t understand what’s meant by that.
In: 5
Vectors are first taught in a geometric setting: it’s a thing which has a specific direction, and a specific length (magnitude), but not a specific location. The vector from (0,4) to (7,5) would be the same as the one from (10,4) to (17,5) for example.
Then, it turns out you can add vectors (by joining them tip to tail) and multiply them by real numbers (by stretching them)
A deeper understanding would be that “vectors” are anything that you can add to each other, and multiply by numbers^(that don’t have to be real numbers any more), as long as this multiplication and addition obeys certain rules. Including a lot of things that don’t seem to have a “direction”. For example, the set of all functions forms a “space” of vectors – you can add functions together, and multiply them by numbers, and so anything you learn about vectors also applies to functions.
I, personally, don’t like the notation for vectors that uses the unit vectors i, j and k. It seems like a lot of wasted ink to write 5i + 6j + 7k, when I could just write (5,6,7), especially once I ink in all the tildes underneath and the little caps on top, that I can’t type here. However, when people are first being introduced to vectors, it can be helpful to
* NOT use a notation that makes them look like points: vectors and points are, strictly speaking, different things, and it can be confusing to write (5,6,7) to mean both the vector 5i+6j+7k and ALSO the point at the tip of that vector when we put the tail at the origin.
* NOT leap too quickly from the geometric “vectors have direction and magnitude” setting to the more abstract “vectors are things that add and multiply like vectors” setting.
For your other questions:
* In general, a “unit vector” is just a vector with length 1.
* Those i, j and k are just specific unit vectors, in the directions of the x-, y- and z-axes.
* You *can* use a unit vector to give the direction of some other vector, but unit vectors (vectors with length 1) are useful for other things too.
* A “normal vector” is a vector “at right angles”. You should ask “at right angles to what?”, but then the answer is “well, normal to what?”. A vector can’t just be “normal”, it has to be at right angles ***to*** something.
* If you’re solving a question about a plane in 3D, space, then a “normal” vector *to the plane* would be a vector at right angles *to the plane*.
* Normal vectors don’t have to be unit vectors. If you specifically want them to be, you should ask for a “unit normal vector”.
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