Using Vectors

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.

Parametric Representation of Lines

  • 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.

Position Vectors

  • 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

  • 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 ).

Vector Applications In Mechanics

  • 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 Proof

  • 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).