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 + λ→b, 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 2ˆi–ˆj+4ˆk and is in the direction ˆi+ˆj–2ˆk.
Solution : Here →a = 2ˆi–ˆj+4ˆk and →b = ˆi+ˆj–2ˆk.
So, the vector equation of the required line is
→r = →a + λ→b
or, →r = (2ˆi–ˆj+4ˆk) + λ(ˆi+ˆj–2ˆk), 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 (→b–→a), 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 (→b–→a)
Here →a = 3ˆi+4ˆj–7ˆk and →b = ˆi–ˆj+6ˆk.
So, the vector equation of the required line is
→r = (3ˆi+4ˆj–7ˆk) + λ (ˆi–ˆj+6ˆk – 3ˆi+4ˆj–7ˆk)
or, →r = (3ˆi+4ˆj–7ˆk) + λ (−2ˆi–5ˆj+13ˆk)
where λ is a scalar.