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
When we’re using vectors in normal, 3-dimensional geometry, i, j, and k are typically used to represent unit (ie, length 1) vectors in the positive direction along the x, y, and z axes, respectively.
So 5i + 6j +7k is the vector that points from the origin, (0, 0, 0), to the point (5, 6, 7).
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”.
Vectors in general are used to tell direction + magnitude. Lets say you’re standing on a 2D grid at the origin (0,0) where every space is 1 step away. You move over to (5, 6) Your X position moved 5 spaces over and your Y position moved 6 spaces up. You traveled a straight line to that spot though, not 5 steps over then 6 steps up. To get the total distance you traveled you can use the pythagorean theorem to get 7.81 steps. This is the magnitude.
A unit vector is a more specific type of vector. It’s the vector you use when you only care about the direction because it’s magnitude is always 1. Why learn all this crazy math to convert a vector into a unit vector? Well because sometimes we only care about the direction.
Going back to our earlier example. What if someone said “I want to know how to get to a position in the same direction as someone traveling from (0,0) to (5,6), but I want to go 20 steps instead of 7.81.” First you find the unit vector of 5i + 6k, ( 0.64i + 0.77k) and then multiply it by 20 (12.8i + 15.4k).
“I want to go where that ship is heading, but twice as fast”
“I want to compute wind drag on a car as it moves across the map. I know how to compute the magnitude of the drag, I just want to know the unit vector so I can set drag to the opposite direction * magnitude of drag.”
There are plenty of uses for unit vectors.
You can scale groups of vectors up and down and their lengths maintain the same ratio to each other. Rather than specifying a quantity for each vector each time you solve a problem, you can just presume a unit vector length of 1, do the math you want using ratios, then apply these to the quantities in the problem at hand.
If we were to “see” a vector and want to describe which way it’s heading to someone, we might be like “umm, it’s pointing *that direction*”, while pointing in that direction. And *that direction* is vague, isn’t it? People do it all the time – “He drove off that way, and he was going about 50 mph”. This is meaningless if you can’t see where I’m pointing when I say *that way*.
Mathematicians, engineers and physicists are not big fans of vague. So, unit vectors provide us with a concrete way of saying *that way*. The 50 mph is important (magnitude) and the unit vector in that direction tells us the rest
Like someone else said, i, j and k aren’t the only unit vectors, but when combined, they form a pretty convenient way to accomplish all this. And not to drift off topic, but that’s what we mean when we say they form a **basis** for 3-dimensional space – you can describe any direction with these vectors and a little multiplication to scale the result down to a magnitude of 1.
Through this, we can all be on the same page with *that way*.
Vectors can be used as a way to ‘define’ space. For example, pick a corner in your room; that is the ‘origin’. You have two directions along the floor and one up along the walls. These are your ‘base’ vectors (let’s say **i**, **j**, **k**, respectively).
When you state the position **p** of an object in the room, they will be in terms of steps along those vectors; say, 5 steps along **i**, 6 along **j** and 7 along **k**. In short: **p** = 5**i** + 6**j** + 7**k**. That’s what that notation means.
The (5, 6, 7) you often see are just the coordinates: scalars for a certain set of vectors that define your space. Now usually, there’s a set of ‘default’ vectors (like x and y axis on a graph) so you don’t need to mention the vectors explicitly, but this is not always true. In the room example, you could have used a different corner. Or you could have used vectors that are not at 90° angles (like following a diamond-shaped floor pattern). In that case, you’d have a different set of base vectors: **i’**, **j’**, **k**’. If you follow the (5, 6, 7) along those vectors, you end up in a different place. If you want to be sure you’re not misunderstood, you need both the vectors and coordinates.
As for unit vectors: so far I’ve only really mentioned the direction of the vectors. But vectors have both a direction and magnitude. When I said ‘in terms of steps along those vectors’, it’s important to know how long those vectors are: if you used ‘1 meter’ long vectors, (5, 6, 7) would again end you in a different spot than if you used ‘1 foot’ long vectors.
A unit vector is simply a vector with a length of 1. When you do math with vectors, their length will often come up as a multiplier. Since 1*x is just x, it’s a very convenient multiplier. Effectively, you’ve taken the ‘magnitude’ out of the equation, so you’re just left with the ‘direction’ part.
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