Using Vectors

Vector Equations

A vector equation can depict a wide variety of mathematical ideas, from simple geometric relationships to complex dynamic systems.
The vector equation of a line in three dimensions is generally expressed as r = a + tb, where a is a position vector, b is a direction vector, t is a scalar parameter.
A line in 2D space can be represented by a similar equation, though it will only involve two dimensions.
Additionally, the vector and scalar (dot) product can be used to derive equations such as the equation of a plane.

The parametric representation of lines offers a way for the equation of a line to be written using a single parameter, often denoted by t.
In a parametric equation, each variable (i.e., x, y, z) is written as a function of this singular parameter.
The parameter t can represent various quantities, such as time, in real-world applications. It provides a way to describe every possible position on the line.
If the position vector of a given point and the direction vector of the line are known, the vector equation of the line can be used to find the parametric form.

A position vector identifies a point in a particular space relative to an origin.
The position vector of a point A is generally represented as OA, where O represents the origin.
It is a vector that starts from the origin of the vector space and points to the coordinates of the point.
Comparing position vectors—e.g., by subtraction—can provide information about the vector connecting two points.

The scalar product (also known as the dot product) is a key operation in vector calculus.
The scalar product of two vectors, a and b, can be found using the formula a•b = a b cos θ, where a and b are magnitudes of the vectors and θ is the angle between them.
Alternatively, in Cartesian form, the scalar product of a = ai + bj and b = ci + dj is simply ac + bd.
The scalar product is commutative and distributive: a•b = b•a, and a•(b+c) = a•b + a•c.
An important use of the scalar product is finding the angle between two vectors, since cos θ = a•b / ( a b ).

The concept of force is often represented using vectors with the magnitude representing the strength of the force and the direction indicating the direction of the force.
Displacement, velocity, and acceleration are other key physical quantities that can be expressed as vectors.
When analysing motion in mechanics, it is often useful to decompose vectors into components along perpendicular axes. Then, the equations of motion can be applied separately along each axis.
The concept of equilibrium states that for a body to be in equilibrium, the vector sum of all forces acting on it must be equal to zero.

Vector methods can be used to prove geometric relationships, such as proving lines are parallel or showing points are collinear.
For example, if the vectors AB and CD are parallel, then AB = kCD for some scalar k.
To show three points are collinear, you could demonstrate one of the connecting vectors is a scalar multiple of the other, such as BC = kAB for some scalar k.
Vectors can also prove ratios, such as showing four points divide a line in a given ratio (e.g., 1:2:3).