Scaling a vector is a pretty neat trick. We can add half an arrow, or double an arrow, or use 0-1 to slide along the length of an arrow. But that trick can’t give us distances.
If we want to travel 5 units along an arrow, or move along it at 2 units/second, we
need more math.
The first trick is getting the length of an arrow. The second is getting a length 1 version of an arrow. After a bit, we’ll start thinking of any arrow as really being those two parts – the direction, and how far.
The basic way to find the distance between two things is to make an arrow between them, then measure the length. So all distances are really measuring how long an arrow is, which is officially called its magnitude.
As a shortcut, Unity lets us measure distance either way: length of an arrow, or distance between two points. Vector3.Distance(A,B) or C.magnitude (no parens, just because.) Examples:
The 1-line call to Distance looks nicer since it’s a shortcut. Inside, it’s
subtracting the points and running magnitude.
We don’t have to use these to measure between objects. They work on any points or arrows, even ones we just make. You might remember this from the pythagorean theorem (a right triangle with sides 3 and 4 has hypotenuse 5):
A’s not really an arrow – we didn’t subtract 2 points to get it. Maybe we intend to
use it as one …. But either way magnitude gives the length as if it were an
Here are some fun facts about distance and magnitude:
Often we don’t need an arrow going all the way to the target. We need one pointing to the target, which we often call a direction arrow. To simplify, direction arrows are usually length 1. For example, transform.forward is our forward direction arrow.
The important thing is that direction arrows only tell us which way to go. The
length is unimportant. For example (2,1,0) and (4,2,0) are the same direction. If you
wanted to write that direction (twice as far x as y) the official way, you should use
trig to make it length 1, but don’t have to.
Raycasts are a nice example of using direction arrows. They take a position and a direction and walk that way until they hit something. This shoots an imaginary ray north from us:
Raycasts think the second input is a direction. (0,0,10) is the forward, +z direction. It would say we were blocked if anything was 1 in front of us, or 10 or any distance. We could have used (0,0,1) for the direction and it would run the same.
If we wanted to check for obstacles sideways and up, dir could be (2,1,0) or
For a comparison, Debug.DrawRay takes an actual offset. It starts at the point you give it, then adds the offset and draws exactly that arrow. In this case, the 10 really means 10:
Having Raycast and DrawRay work differently is for sure confusing. But it does a nice job of showing the terms: offset means we care about the whole arrow and where the tip ends, and direction means we don’t.
(Funny story: in the manual DrawRay says it takes a direction, but that’s a typo. It’s an offset.)
You can often use a direction arrow of any length, but there are some tricks you can do with an arrow of exactly length 1. The math term for that is a normalized direction, or sometimes a unit vector (which is shorthand for “a 1-unit long vector.”)
There’s a really slick trick to turn an arrow into length 1: divide by its length. Here’s an example getting a length 1 direction to a marker:
This trick works for any arrow – it can be pointing backwards, or have length less
than 1 (it will grow) – and it still works.
Unity provides two shortcut functions for that. One of them makes you be length one, and another makes a length one version of you. Examples:
Normalizing is the ten dollar math term for getting a length 1 version of an arrow. But it’s nothing special besides dividing by the length. For example, (1,0,0) is normalized, which is a fancy way of saying it’s already length 1.
Some normalizing notes:
With the theory out of the way, we’re ready to do tricks by breaking an offset into length one direction and magnitude. We start with this:
Now toB is a length 1 direction arrow towards B, and len is how far.
The simplest trick is that A+toB is one unit from A towards B. A+toB*3.5f is
exactly 3.5 units from A towards B. We can pick the exact distance in A+toB*dist.
We can slide dist from 0 to the total, len, to walk a real distance from A to
Here are few simple examples. This puts a “shield” two units away from us, facing the marker:
You might remember from before we could use a fraction of a vector
to do something similar. The improvement here is we can give an actual
Our old cube-hiding was behind us 1/10th of the distance to what it was hiding from. That made it closer or further, depending. Not what we wanted. Distance math lets it always hide 3 units behind:
This next example slides a ball from us to a marker. We did that before, but now it moves at a constant rate (before, it took about 2 seconds, no matter how close or far we were):
This way looks better for most things – moving twice as far finally takes twice as
long. The percent method is nice for displays – faster movement gives a hint that the
target is further away.
The unit vector trick is also great for throwing a rigidbody at something. Before, we could shoot a ball at 10 units/second in our forward direction; but not towards a target. Now we can.
This fires a ball at a marker, always with a speed of 5:
Notice how it finds the total arrow from the ball to the marker, not from the player. Otherwise the angle might be off. We don’t bother computing the distance to the marker, since we didn’t need it.
Sometimes you look at the numbers for distance, and they seem funny. If you never
do, skip this.
A surprising thing is, when you have a long length and a short one, the
short one counts for almost nothing. For example (10,1,0), has a length of
only 10.05. Going up by one added just 0.05 to the distance. If you flip it
around, this makes sense: imagine driving to a town 10 miles east and 1 mile
north. That’s 11 miles if we have to drive that way. But we know a diagonal
straight-shot road will be a good deal – it will be just a little longer than 10
Even when the numbers are close together, the answer is smaller than it seems.
(5,4,0) has a length of 6.4. In 3D, the numbers are even shorter. The arrow (3,4,5)
has a length of only 7.1. The answer had to be at least 5, and the 3 and 4 didn’t add
If you estimate distance between two points in your head, you can think of all
differences as positive. For example, comparing (10,10,10) to (2,13,9). All that
matters is: 8 away, 3 away and 1 away. So it’s like an arrow (8,3,1). The distance will
be 8 plus a little more.
For real, distances are computed using the Pythagorean theorem: x2 + y2 = d2.
That’s why (3,4,0) has length 5.
Normalized (length one) vectors have the same funny-looking math as
transform.forward. A unit diagonal arrow really is (0.71, 0.71, 0). If you normalize (1,1,0), that’s what you get. A unit arrow at 30 degrees really is (0.6, 0, 0.8). Almost all unit arrows are wrong-looking numbers like that.
But, to repeat myself, you don’t need to know these numbers. If you know trig or want to learn, it’s fun to look at them. If you notice (0.6, 0.8, 0) and think “wait, aren’t they suppose to be length 1?” now you know they don’t add to one – they pythagorean square add to 1.