This is a vector:
A vector has magnitude (how long it is) and direction:
The length of the line shows its magnitude and the arrowhead points in the direction.
We can add two vectors by simply joining them head-to-tail:
And it doesn't matter which order we add them, we get the same result:
Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.
The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.
If you watched the plane from the ground it would seem to be slipping sideways a little.
Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that.
We can also subtract one vector from another:
- first we reverse the direction of the vector we want to subtract,
- then add them as usual:
A vector is often written in bold, like a or b.
|A vector can also be written as the letters
of its head and tail with an arrow above it, like this:
Now ... how do we do the calculations?
The most common way is to break up a vector into x and y pieces, like this:
The vector a is broken up into
the two vectors ax and ay
And we can add the vectors by adding the x parts then adding the y parts:
The vector (8,13) and the vector (26,7) add up to the vector (34,20)
Example: add the vectors a = (8,13) and b = (26,7)
c = a + b
c = (8,13) + (26,7) = (8+26,13+7) = (34,20)
Remember: to subtract, first reverse the vector we want to subtract, then add.
Example: subtract k = (4,5) from v = (12,2)
a = v + −k
a = (12,2) + −(4,5) = (12,2) + (−4,−5) = (12−4,2−5) = (8,−3)
Magnitude of a Vector
The magnitude of a vector is shown by two vertical bars on either side of the vector:
OR it can be written with double vertical bars (so as not to confuse it with absolute value):
We use Pythagoras' theorem to calculate it:
|a| = √( x2 + y2 )
Example: what is the magnitude of the vector b = (6,8) ?
|b| = √( 62 + 82 ) = √( 36+64 ) = √100 = 10
A vector with magnitude 1 is called a Unit Vector.
Vector vs Scalar
When using vectors we call an ordinary number a "scalar".
Scalar: just a number (like 7 or −0.32) ... definitely not a vector.
A vector is often written in bold, so we know it is not a scalar:
- so c is a vector, it has magnitude and direction
- but c is just a value, like 3 or 12.4
Example: kb is actually the scalar k times the vector b.
Multiplying a Vector by a Scalar
When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.
Example: multiply the vector m = (7,3) by the scalar 3
|a = 3m = (3×7,3×3) = (21,9)|
It still points in the same direction, but is 3 times longer
(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)
Multiplying a Vector by a Vector (Dot Product and Cross Product)
How do we multiply two vectors together? There is more than one way!
(Read those pages for more details.)
More Than 2 Dimensions
The vectors we have been looking at have been 2 dimensional, but vectors work perfectly well in 3 or more dimensions:
Example: add the vectors a = (3,7,4) and b = (2,9,11)
c = a + b
c = (3,7,4) + (2,9,11) = (3+2,7+9,4+11) = (5,16,15)
Example: subtract (1,2,3,4) from (3,3,3,3)
(3,3,3,3) + −(1,2,3,4)
= (3,3,3,3) + (−1,−2,−3,−4)
Example: what is the magnitude of the vector w = (1,−2,3) ?
|w| = √( 12 + (−2)2 + 32 ) = √( 1+4+9 ) = √14
Magnitude and Direction
We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):
|Vector a in Polar
|Vector a in Cartesian
You can read how to convert them at Polar and Cartesian Coordinates, but here is a quick summary:
|From Polar Coordinates (r,θ)
to Cartesian Coordinates (x,y)
|From Cartesian Coordinates (x,y)
to Polar Coordinates (r,θ)
Sam and Alex are pulling a box.
What is the combined force, and its direction?
Let us add the two vectors head to tail:
Now, convert from polar to Cartesian (to 2 decimals):
- x = r × cos( θ ) = 200 × cos(60°) = 200 × 0.5 = 100
- y = r × sin( θ ) = 200 × sin(60°) = 200 × 0.8660 = 173.21
- x = r × cos( θ ) = 120 × cos(-45°) = 120 × 0.7071 = 84.85
- y = r × sin( θ ) = 120 × sin(-45°) = 120 × -0.7071 = −84.85
Now we have:
Now it is easy to add them:
(100, 173.21) + (84.85, −84.85) = (184.85, 88.36)
We can convert that to polar for a final answer:
- r = √ ( x2 + y2 ) = √ ( 184.852 + 88.362 ) = 204.88
- θ = tan-1 ( y / x ) = tan-1 ( 88.36 / 184.85 ) = 25.5°
And we have this (rounded) result:
And it looks like this for Sam and Alex:
They might get a better result if they were shoulder-to-shoulder!