Vector equation of a line

Vector Equation of a Line

Understanding the Vector Equation of a Line

• A central concept in Core Pure Maths is the vector equation of a line in two or three dimensions.
• A vector equation represents a line as a direction vector and a position vector.
• The position vector, often represented as ‘a’, is the vector from the origin to any point on the line.
• The direction vector, often represented as ‘b’, signifies the direction in which the line runs.

Writing the Vector Equation of a Line

• The vector equation of a line can be written in the format r = a + λb.
• Here, ‘r’ represents any vector on the line, ‘λ’ is a scalar parameter which modifies the direction vector ‘b’.
• Any value of λ will give the position vector of some point on the line.

Identifying Position and Direction Vectors

• To find the position vector ‘a’ in the equation, choose a point on the line and find its coordinates. This vector from the origin to the point forms the position vector.
• The direction vector ‘b’ can be obtained by subtracting the coordinates of one point on the line from another.

Interpreting the Vector Equation of a Line

• Each possible value of the parameter λ corresponds to exactly one point on the line.
• By varying λ, we can generate all the points on the line – positive values of λ generate points in the direction of ‘b’, while negative values generate points in the opposite direction.
• The direction vector ‘b’ is very important: it indicates the slope and direction of the line.

Uses of the Vector Equation of a Line

• Vector equations of lines are essential tools in physics for modelling trajectories, in computer graphics for rendering lines and curves, and in engineering for designing structures.

Remember to practice writing and interpreting vector equations of a line using various points and direction vectors to solidify your understanding of these concepts.