Vector Equation of a line and a Plane

Vector equations are used to represent lines or planes in a three-dimensional framework, and they are also known as vector algebra. When considering a three-dimensional plane, three coordinates with respect to three axes are required, and here vectors are useful in representing the vector equation of a line or plane more easily than they would be otherwise. In a three-dimensional framework, the unit vector along the x-axis is denoted by the letter i∧ the unit vector along the y-axis is denoted by the letter j∧, and the unit vector along the z-axis is denoted by the letter k∧. The vector equations are written in the three-dimensional plane using the symbols i∧ j∧, and k∧, and they can be represented geometrically in the three-dimensional plane. The simplest form of the vector equation of a line is r→ = a + λb→, and the simplest form of the vector equation of a plane is r→. n∧ = d. The vector equation of a line is represented by the symbol r→.

Vector Equation of a Line: r→ = a + λb→

Vector Equation of a Plane:  r→. n∧ = d

In order to find the vector equations of a line, two methods are used, and four methods are used in order to find the vector equations of a plane.

Vector equation of a line

It is possible to compute vector equations for a line by using any two points on the line, or by using a point on the line and a parallel vector. The following are the two methods for forming a vector form of the equation of a line in vector form:

• In mathematics, the vector equation of a line passing through a point and having a position vector a, while also being parallel to a vector line b, is denoted by the notation r→ = a + λb→ .

• This is the vector equation of a line passing through two points with the position vector a and the position vector b: r→ = a→ + λ(b→ – a→).

Vector equation of a plane

The vector equation of a plane is a vector form of the equation of a plane in a cartesian coordinate system, and it can be computed using a variety of methods depending on the values of the plane that are available as inputs. The four different expressions for the equation of a plane in vector form are shown in the following list.

• Normal Form: The equation of a plane located at a perpendicular distance d from the origin and having a unit normal vector n is denoted by the symbol R. The normal vector n equals the distance between the origin and the plane.
• Through a point and perpendicular to a given Line, we have In the case of a plane that is perpendicular to one of the two given vectors N→ and passes through point a→, the equation is (r→₋a→).N→=0.

• Through three non-collinear lines: It is equal to zero when a plane passes through three non-collinear points (e.g., a→, b→, and c→) is (r→₋a→) [ (b→₋a→) ×  ( c→₋a→)] =0.
• The intersection of two planes: It is represented by the symbol When a plane passes through the intersection of two planes, r→.n∧1 = d1 and r→.n∧ = d2, the equation for the plane passing through the intersection is (r→.n∧1 + λn∧2) = d1 +λ d2.

Example:

1 Find the vector equation of the line that passes through the points (3, 5, -2), and that is parallel to the vectors 5 i∧ + j∧ + 4 k∧.

Solution  The given point is (3, 5, -2), and the given vector is (5 i∧ +  j∧ + 4k∧).

a = 3i∧ + 5j∧ – 2k∧ and b = 5i∧ + j∧ + 4k∧ are two ways to represent these functions, respectively.

The vector equation of a line passing through a point id a and parallel to a vector line b→ is given by r→ = a→ + λb→.

Conclusion:

Vector equations are used to represent the equation of a line or a plane with the help of the variables x, y, and z in order to simplify the equation. Vector equations are used to represent lines or planes in a three-dimensional framework, and they are also known as vector algebra. The vector equation specifies where a line or a plane should be placed in a three-dimensional space using a mathematical formula. 

In a three-dimensional framework, the unit vector along the x-axis is denoted by the letter I∧ the unit vector along the y-axis is denoted by the letter j∧, and the unit vector along the z-axis is denoted by the letter k∧.

In mathematics, the vector equation of a line passing through a point and having a position vector a, while also being parallel to a vector line b, is denoted by the notation r→ = a + λb→.

This is the vector equation of a line passing through two points with the position vector a and the position vector b: r→ = a→ + λ(b→ – a→).