Vector Form of Equation of Line in 3D

Vector Form of the Equation of Line in 3D is given using a vector equation that involves the position vector of the points. In this heading, we will obtain the 3D Equation of the line in vector form for two cases.

Case 1: 3D Equation of Line Passing through Two Points in Vector Form

Let us assume we have two points A and B whose position vector is given as [Tex]\vec a[/Tex] and [Tex]\vec b[/Tex].

Then the vector equation of the Line L is given as

[Tex]\vec l = \vec a + \lambda (\vec b – \vec a) [/Tex]

where [Tex](\vec b – \vec a)[/Tex] is the distance between two points and λ is the parameter that lies on the line.

Derivation of 3D Equation of Line Passing through Two Points in Vector Form

Suppose we have two points A and B whose position vector is given as [Tex]\vec a[/Tex] and [Tex]\vec b[/Tex]. Now we know that a line is the distance between any two points. Hence, we need to subtract the two position vectors to obtain the distance.

⇒ [Tex]\vec d = \vec b – \vec a [/Tex]

Now we know that any point on this line will be given as the sum of position vector [Tex]\vec a \space or \space \vec b [/Tex] with the product of the parameter λ and the position vector of the distance between two points i.e. [Tex]\vec d [/Tex]

Hence, the equation of the line in the vector form will be [Tex]\vec l = \vec a + \lambda (\vec b – \vec a)[/Tex] or [Tex]\vec l = \vec b + \lambda (\vec a – \vec b)[/Tex]

Example: Find the vector equation of a line in 3D that passes through two points whose position vectors are given as 2i + j – k and 3i + 4j + k

Solution:

Given that the two position vectors are given as 2i + j – k and 3i + 4j + k

Distance d = (3i + 4j + k) – (2i + j -k) = i + 3j + 2k

We know that equation of the line is given as [Tex]\vec l = \vec a + \lambda (\vec b – \vec a) [/Tex]

Hence, the equation of the line will be [Tex]\vec l[/Tex] = 2i + j – k + λ(i + 3j + 2k)

Case 2: Vector Form of 3D Equation of Line Passing through a Point and Parallel to a Vector

Let’s say we have a point P whose position vector is given as [Tex]\vec p[/Tex]. Let this line be parallel to another line whose position vector is given as [Tex]\vec d [/Tex].

Then the vector equation of the line ‘l’ is given as

[Tex]\vec l = \vec p + \lambda \vec d[/Tex]

where λ is the parameter that lies on the line.

Derivation of the Vector Form of 3D Equation of Line Passing through a Point and Parallel to a Vector

Consider a point P whose position vector is given as [Tex]\vec p[/Tex]. Now let us assume this line is parallel to a vector [Tex]\vec d[/Tex] then, the equation of the line will be [Tex]\vec l = \lambda \vec d[/Tex]. Now since the line also passes through point P, then as we move away from Point P in either direction on the line then the position vector of the point will be in the form of [Tex]\vec p + \lambda \vec d [/Tex]. Hence, the equation of the line will be [Tex]\vec l = \vec p + \lambda \vec d[/Tex] where λ is the parameter that lies on the line.

Example: Find the Vector Form of the Equation of the line passing through the point (-1, 3, 2) and parallel to a vector 5i + 7j – 3k.

Solution:

We know that the vector form of the equation of a line passing through a point and parallel to a vector is given as [Tex]\vec l = \vec p + \lambda \vec d[/Tex]

Given that the point is (-1, 3, 2), hence the position vector of the point will be -i + 3j + 2k and the given vector is 5i + 7j – 3k.

Therefore, the required equation of the line will be [Tex]\vec l [/Tex] = (-i + 3j + 2k) + λ(5i + 7j – 3k).

The equation of a line in a plane is given as y = mx + C where x and y are the coordinates of the plane, m is the slope of the line and C is the intercept. However, the construction of a line is not limited to a plane only.

We know that a line is a path between two points. These two points can be located anywhere whether they could be in a single plane or they could be in space. In the case of a plane, the location of the line is characterized by two coordinates arranged in an ordered pair given as (x, y) whereas in the case of space, the location of the point is characterized by three coordinates expressed as (x, y, z).

In this article, we will learn the different forms of equations of lines in 3D space.