Topics
Vector arithmetic & geometry
0 of 0 exercises completed
Addition, subtraction and scalar multiplication, computed algebraically and geometrically
Want a deeper conceptual understanding? Try our interactive lesson!
2D vector notation
AHL AI 3.10
Just as points in the plane have an x and a y, 2D vectors have 2 components. We call these i and j.
The following diagram shows the vector −3i−4j, which can also be written as (−3−4)
Concept of a vector
AHL AI 3.10
A vector is an arrow in space. It specifies movement in a particular direction. For example, the vector i+2j means "move 1 unit right and 2 units up".
The difference between the vector i+2j and the point (1,2) is that a vector can start anywhere, and only specifies a movement. For example, all the vectors below are i+2j.
Addition of 2D vectors
AHL AI 3.10
Vectors in 2D can be added an subtracted using a diagram. For example, if a=−i+2j and b=3i−2j, then we can find 2a+b as follows:
So 2a+b=i+2j, which we can also find by adding components:
2a+b=2(−i+2j)+(3i−2j)=(−2+3)i+(4−2)j
Position Vectors
AHL AI 3.10
Vectors can be used to describe the position of points in space, relative to the origin O. For example, if the point A has coordinates A(3,1), then its position vector is
OA=3i+j=(31)
This vector tells you how to get from the origin to point A, and is only subtly different from the concept of coordinates. If B has coordinates B(−4,3), then its position vector is
OB=−4i+3j=(−43)
In the diagram below, the arrows are the position vectors, which are technically different from the points themselves.
Displacement vectors and triangle law
AHL AI 3.10
A displacement vector tells you how to go from one point to another in space.
For example, suppose point A has position vector OA=3i+j=(31), and B has position vector OB=−4i+3j=(−43).
The displacement vector AB is the vector that goes from A to B:
To find the displacement vector AB, we could count squares on the grid. But the more important thing to realize is that these three vectors form a triangle, and if we reverse one of the position vectors it forms a loop, meaning we've gone nowhere:
OA+AB+BO=(00)
But
BO=−OB=(4−3)
So
(31)+AB+(4−3)=(00)
Now isolate:
AB=(00)−(31)−(4−3)=(−72)
In general, we find the displacement vector from X to Y by finding
XY=OY−OX
You can think of this as saying
to get from X to Y, first go from X to O, and then from O to Y
Scalar multiples of vectors
AHL AI 3.10
Scaling a vector means multiplying it by a number k, which stretches the vector by a factor of k. If k<0, then the direction of the vector flips.
Parallel vectors
AHL AI 3.10
Two vectors are parallel if one can be written as a scalar multiple of the other, i.e., if u=kv for some scalar k. Parallel vectors have identical or exactly opposite directions.
3D base and column vectors
AHL AI 3.10
A vector is a mathematical object that has both magnitude (size) and direction in space. A three-dimensional vector is typically described by its components along the x, y, and z axes.
In 3 dimensions, a vector is expressed as
⎝⎛v1v2v3⎠⎞=v1i+v2j+v3k
where i, j, and k are called the base vectors:
i=⎝⎛100⎠⎞,j=⎝⎛010⎠⎞,k=⎝⎛001⎠⎞
Adding & subtracting vectors in 3D
AHL AI 3.10
To add two vectors, add their corresponding components:
⎝⎛x1y1z1⎠⎞+⎝⎛x2y2z2⎠⎞=⎝⎛x1+x2y1+y2z1+z2⎠⎞
To subtract two vectors, subtract their corresponding components:
⎝⎛x1y1z1⎠⎞−⎝⎛x2y2z2⎠⎞=⎝⎛x1−x2y1−y2z1−z2⎠⎞
Addition corresponds to placing vectors head-to-tail, and subtraction corresponds to the displacement between their tips.
Zero vector and negative vector
AHL AI 3.10
The zero vector 0 is a special vector without size or a defined direction, represented by
0=⎝⎛000⎠⎞
The negative of a vector reverses the vector's direction while maintaining its size. If v=⎝⎛xyz⎠⎞, then the negative is
−v=⎝⎛−x−y−z⎠⎞
pointing in exactly the opposite direction.
Vector Magnitude
AHL AI 3.10
The magnitude (or length) of a vector v=⎝⎛v1v2v3⎠⎞ is calculated as
∣v∣=√v12+v22+v32📖
Geometrically, this represents the distance from the vector's starting point to its endpoint in 3D space. For example, the magnitude of the displacement vector AB equals the straight-line distance between points A and B.
Unit vectors
AHL AI 3.10
When a vector u has a magnitude of 1, we say u is a unit vector.
We often express other vectors v with magnitude ∣v∣=1 as scalar multiples of the unit vector pointing in the same direction,
v=ku
where k=∣v∣.
The unit vector in the same direction as a non-unit vector v is written as
u=∣v∣v
Rescaling to form a vector with a magnitude of one is referred to as normalizing a vector.
Nice work completing Vector arithmetic & geometry, here's a quick recap of what we covered:
Skills covered
Exercises checked off
I'm Plex, here to help you understand this concept!