13 B Lines in 2D and 3D

The vector AB and the vector equation of the line AB are very different things. x x x x The line AB is a line passing through the points A and B and has infinite length. The vector AB The line through A and B. The vector AB has a definite length ( magnitude ).  Vector Equation of a Line

Finding the Equation of a Line In coordinate geometry, the equation of a line is e.g. The equation gives the value (coordinate) of y for any point which lies on the line. The vector equation of a line must give us the position vector of any point on the line. We start with fixing a line in space. We can do this by fixing 2 points, A and B. There is only one line passing through these points. Vector Equation of a Line

In Other Words: We can determine the vector equation of a line using its direction and any fixed point on the line.

x x x We consider several more points on the line. x x x A and B are fixed points. We need an equation for r, the position vector of any point R on the line. r1r1 a Starting with R 1 : Vector Equation of a Line

x x x We consider several more points on the line. x x x x a r2r2 A and B are fixed points. Starting with R 1 : We need an equation for r, the position vector of any point R on the line. Vector Equation of a Line

x x x We consider several more points on the line. x x x x a r3r3 A and B are fixed points. Starting with R 1 : We need an equation for r, the position vector of any point R on the line. Vector Equation of a Line

x x x x x x x a So for R 1, R 2 and R 3 For any position of R, we have t is called a parameter and can have any real value. It is a scalar not a vector. Vector Equation of a Line

x x x x x x x a This is what I’m Saying: t is called a parameter and can have any real value. It is a scalar not a vector. Fixed Point Any Point b Say you have a line that passes through a fixed point A with position vector a, and that line is parallel to vector b. Point R is on the line r By vector addition: Vector Equation of a Line

x x x x x x x a Fixed Point Any Point b Since is parallel to b, r Vector Equation of a Line

direction vector... SUMMARY  The vector equation of the line through 2 fixed points A and B is given by  The vector equation of the line through 1 fixed point A and parallel to the vector is given by position vector... of a known point on the line of the line Vector Equation of a Line

In 2-D, we are dealing with a line in a plane A (a 1, a 2 ) R (x, y) The vector equation of the line Is the direction vector Slope or gradient is:

We can write the parametric equations of the line: Each point on the line corresponds to exactly one value of t. Convert into Cartesian form by equating t values. We get the Cartesian equation of the line as:

A line passes through the point A(1,5) and has the direction vector. Describe the line using: Draw a sketch A (1, 5) R (x, y) O r a A) A vector equation

A line passes through the point A(1,5) and has the direction vector. Describe the line using: A (1, 5) R (x, y) O r a B) Parametric Equations

A line passes through the point A(1,5) and has the direction vector. Describe the line using: A (1, 5) R (x, y) O r a C) Cartesian Equation

In 3-D, we are dealing with a line in Space A (a 1,a 2, a 3 ) R (x,y,z) Is the vector equation of the line Is the direction vector Slope or gradient NONE. We just describe its direction by its direction vector

We can write the parametric equations of the line: Each point on the line corresponds to exactly one value of t.

Find a vector equation and the corresponding parametric equations of the line through (1,-2, 3) in the direction of 4i + 5j – 6k The parametric equations are: The vector equation is: r = a + tb

Non-Uniqueness of a line Consider the line passing through (5,4) and (7,3). When writing the equation of the line, we could use either point to give the position vector a (5,4) (7,3) We could also use the Direction vector We could also use the Direction vector We could also use ANY Direction vector that is a non-zero scalar multiple of these vectors

Non-Uniqueness of a line So, we could write the equation of the line as: (5,4) (7,3) We could keep going.

Non-Uniqueness of a line So, we could write the equation of the line as: (5,4) (7,3)

Non-Uniqueness of a line Find a parametric equations of the line through A(2,-1,4) and B(-1,0,2) We need a direction vector for the line, either Using point A, the equations are: Using point B, the equations are:

Homework Page 325 (1 – 8)