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Vector arithmetic & geometry
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3D and 2D vector notation with base vectors and position vectors, including zero and negative vectors, scalar multiples, unit vectors, vector magnitude, addition and subtraction by components or triangle law, displacement vectors, and parallel vectors.
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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
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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.
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