Chapter 3
Direction & Length

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.

3.1 Magnitude/distance

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:

float dist = Vector3.Distance(transform.position, marker.position);  
print("You are "+dist+" away from the marker");  
// same thing: get arrow to marker, then measure it:  
Vector3 toMarker=marker.position-transform.position; // arrow from us to marker  
dist = toMarker.magnitude; // length of arrow = distance

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):

Vector3 A = new Vector3(3,0,4);  
dist = A.magnitude; // 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 arrow.

Here are some fun facts about distance and magnitude:

3.1.1 Direction

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:

Vector3 dir=new Vector3(0,0,10);  
if(Physics.Raycast(transform.position, dir))  
  print("forward is blocked");

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 (4,2,0).

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:

Vector3 dir=new Vector3(0,0,10);  
Debug.DrawRay(transform.position, dir); // draws length 10 forward arrow  
dir=new Vector3(0,0,0.333f);  
Debug.DrawRay(transform.position, dir); // draws very short forward arrow

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.)

3.1.2 Normalized direction

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:

Vector3 toMarker = marker.position - transform.position;  
float dist = toMarker.magnitude;  
Vector dirToMarker = toMarker/dist; // length 1 arrow to 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:

A = new Vector3(3,4,0);  
A.Normalize(); // A is now (0.6, 0.8, 0), which happens to be length 1  
B = A.normalized; // B is (0.6, 0.8, 0), A is unchanged  
A=A.normalized; // same as A.Normalize();

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:

3.2 Normalized direction + length

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:

Vector3 toB = B-A;  
float len = toB.magnitude;  
toB = toB.normalized;

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 B.

Here are few simple examples. This puts a “shield” two units away from us, facing the marker:

// get length 1 arrow to marker, all in one line:  
Vector3 toMarker = (marker.position-transform.position).normalized;  
shield.LookAt(marker); // not needed, but fun

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 distance.

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:

Vector3 toMarker = (marker.position-transform.position).normalized;  
// hiding from marker, behind us and 3 away:  
hidingCube.position=transform.position - toMarker*3;

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):

public Transform ball;  
float ballDist=0; // this is the actual distance from us, in units  
void Update() {  
  // direction and distance to marker:  
  Vector3 toMarker = marker.position-transform.position;  
  float len=toMarker.magnitude;  
  // slide ballDist from 0 to total len, at 2 units/sec:  
  ballDist+=0.01f; // this many meters per frame  
  if(ballDist>len) ballDist=0;  

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:

if(Input.GetKeyDown(KeyCode.Space)) { // space key fires  
  // start the ball a little ahead of us:  
  Transform ball = Instantiate(ballPrefab);  
  ball.position = transform.position + transform.forward*2  
  Vector3 toMarker=(marker.position-ball.position).normalized;  
  Vector3 vel = toMarker*5; // unit vector times speed  
  ball.GetComponent<Rigidbody>().velocity = vel;  

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.

3.3 Looking at the numbers

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 miles.

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 much.

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.