Loading [MathJax]/jax/output/HTML-CSS/jax.js

Equation of a Line in Vector Form

Here you will learn equation of a line in vector form passing through a fixed point and passing through two points.

Let’s begin –

Equation of a Line in Vector Form

The vector equation of a straight line passing through a fixed point with position vector a and parallel to a given vector b is 

r = a + λb, where λ is scalar.

Note : In the above equation r is the position vector of any point P (x, y, z) on the line. Therefore, r = xˆi+yˆj+zˆk.

Example : Find the vector equation of a line which passes through the point with position vector 2ˆiˆj+4ˆk and is in the direction ˆi+ˆj2ˆk.

Solution : Here a = 2ˆiˆj+4ˆk and b = ˆi+ˆj2ˆk.

So, the vector equation of the required line is

r = a + λb

or, r = (2ˆiˆj+4ˆk) + λ(ˆi+ˆj2ˆk), where λ is a scalar.

Equation of Line in Vector Form Passing Through Two Points

The vector equation of line passing through two points with position vectors a and b is

r = λ (ba), where λ is a scalar

Example : Find the vector equation of a line which passes through the point A (3, 4, -7) and B (1, -1, 6)

Solution : We know that the vector equation of line passing through two points with position vectors a and b is,

r = λ (ba)

Here a = 3ˆi+4ˆj7ˆk and b = ˆiˆj+6ˆk.

So, the vector equation of the required line is

 r = (3ˆi+4ˆj7ˆk) + λ  (ˆiˆj+6ˆk3ˆi+4ˆj7ˆk)

or, r = (3ˆi+4ˆj7ˆk) + λ (2ˆi5ˆj+13ˆk)

where λ is a scalar.

Leave a Comment

Your email address will not be published. Required fields are marked *