Cartesian Form

The cartesian form helps to represent geometric entities in the cartesian plane. A point, a line, or a plane can be easily represented in a three-dimensional plane, across the x-axis, y-axis, z-axis, in cartesian form. The cartesian form of representation of a point is (x, y, z), the line is (x - x₁)/a = (y - y₁)/b = (z - z₁)/c, and the plane is ax + by + cz = d.

The cartesian form is helpful to represent the geometric entities as algebraic expressions in three-dimensional geometry. Let us learn more about the conversion of cartesian form to vector form, the difference between cartesian form and vector form, with the help of examples, FAQs.

The cartesian form helps in representing a point, a line, or a plane in a two-dimensional or a three-dimensional plane. The cartesian form is represented with respect to the three-dimensional cartesian system and is with reference to the x-axis, y-axis, and z-axis respectively. The simplest form of cartesian form of the equation of a line is The vector form of the position vector of point A in the three-dimensional cartesian plane is \(\vec A = x\hat i + y\hat j + z\hat k\), which is also represented in cartesian form as a point A(x, y, z).

Let us check the representation of a point, a line, and a plane in cartesian form.

Representation Of Point In Cartesian Form

The representation of a point in a three-dimensional cartesian plane is (x, y, z), and each of x,y, z represent the coordinates of the points with respect to the x-axis, y-axis, and z-axis respectively.

Representation Of A Line In Cartesian Form

The cartesian form of equation of a line passing through the point \((x_1, y_1, z_1)\), and having the direction cosines a, b, c is \(\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} =\dfrac{z - z_1}{c}\). Also, another cartesian form of equation of a line passing through the points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), is \(\dfrac{x - x_1}{x_2 - x_1} =\dfrac{y - y_1}{y_2 - y_1} = \dfrac{z - z_1}{z_2 - z_1}\).

Representation Of A Plane In Cartesian Form

The cartesian form of the equation of a plane having a, b, c as the direction cosines of the normal to the plane and d as the distance of the plane from the origin is ax + by + cz = d.

The cartesian form of equation of a plane passing through the three points\((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), \((x_3, y_3, z_3)\) is \(\begin{vmatrix}x-x_1&y-y_1&z-z_1\\x_2 - x_1&y_2 - y_1&z_2 - z_1\\x_3 - x_1&y_3 - y_1&z_3 - z_1\end{vmatrix}=0\).

The cartesian form can be easily transformed into vector form, and the same vector form can be transformed back to cartesian form. This can be done using two simple techniques. First, the arbitrary form of vector \(\vec r\) is written as \(\vec r = x\hat i + y\hat j + z\hat k \). Secondly, the formula of the product of unit vectors is helpful in converting the cartesian form to vector form.

\(\hat i .\hat i = \hat j. \hat j = \hat k.\hat k = 1\)

\(\hat i. \hat j = \hat j.\hat k = \hat k. \hat i = 0\)

Let us understand this with the help of a simple conversion of the equation of a line from vector form to cartesian form. The equation of a line passing through two points \(\vec a \), and \(\vec b\), is represented as follows.

\(\vec r = \vec a + λ(\vec b - \vec a)\)

Let us take the arbitrary vector \(\vec r \) as \(\vec r = x\hat i + y\hat j + z\hat k\), the points \(\vec a\) and \(\vec b\) as \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\) respectively. This can be represented in the above vector form to obtain the required cartesian form of equation of a line.

\(x\hat i + y\hat j + z\hat k = x_1\hat i + y_1\hat j + z_1\hat k + λ((x_2 - x_1)\hat i + (y_2 - y_1)\hat j + (z_2 - z_1)\hat k)=0\)

\((x - x_1)\hat i + (y - y_1)\hat j + (z - z_1)\hat k = λ((x_2 - x_1)\hat i + (y_2 - y_1)\hat j + (z_2 - z_1)\hat k)=0\)

\(\dfrac{x - x_1}{x_2 - x_1} =\dfrac{y - y_1}{y_2 - y_1} = \dfrac{z - z_1}{z_2 - z_1}\)

This is the required cartesian form of equation of a line.

The difference between a cartesian form and a vector form can be observed for a point, a line, and a plane. The following points presents the difference between the cartesian form and vector form.

• The cartesian form of representation for a point is A(a, b, c), and the same in vector form is a position vector \(\vec OA = a\hat i + b\hat j + c\hat k\).
• The vector form of the equation of a line is \(\vec r = \vec a + λ\vec b\), and the cartesian form of the equation of a line is \(\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}\).
• The vector form of the equation of a line passing through two points is \(\vec r = \vec a + λ(\vec b - \vec a)\), and the cartesian form of the equation of the line is \(\dfrac{x - x_1}{x_2 - x_1} =\dfrac{y - y_1}{y_2 - y_1} = \dfrac{z - z_1}{z_2 - z_1}\).
• The vector form of the equation of a plane having the normal vector \(\vec n\), and at a distance of d units from the origin is \(\vec r.\vec n = d\), and the cartesian form of the equation of a plane with a, b, c as the direction ratios of the plane, and d as the distance of the plane from the origin is ax + by + cz = d.
• The vector form of the equation of a plane passing through a point \(\vec a\) and is perpendicular to vector \(\vec N \) is \((\vec r - \vec a).\vec N = 0\), and the cartesian form of equation of the plane is \(A(x - x_1) + B(y - y_1) + c(z - z_1) = 0\).
• The vector form of equation of plane passing through three non collinear points \(\vec a\), \(\vec b\), \(\vec c\) is \((\vec r - \vec a)[(\vec b - \vec a) × (\vec c - \vec a)]=0\), and the cartesian form of equation of a line passing through three non collinear points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), \((x_3, y_3, z_3)\) is \(\begin{vmatrix}x-x_1&y-y_1&z-z_1\\x_2 - x_1&y_2 - y_1&z_2 - z_1\\x_3 - x_1&y_3 - y_1&z_3 - z_1\end{vmatrix}=0\).

☛Related Topics

• Cartesian Plane
• Cartesian Product
• Coordinate Geometry
• Polar Coordinates

What Is Cartesian Form?

The cartesian form helps to represent geometric entities in the cartesian plane. A point, a line, or a plane can be easily represented in a three-dimensional plane, across the x-axis, y-axis, z-axis, in cartesian form. The cartesian form of representation of a point is (x, y, z), the line is (x - x₁)/a = (y - y₁)/b = (z - z₁)/c, and the plane is ax + by + cz = d

How To Write In Cartesian Form?

The cartesian form for a point can be written as (x, y, z), and each of these coordinates represent with respect to the x-axis, y-axis, and z-axis respectively. the cartesian form for a line can be written as \(\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}\), and the cartesian form for a plane is written as ax + by + cz = d.

How To Convert Vector Form To Cartesian Form?

The vector form can be easily converted to cartesian form by representing the arbitrary vector \(\vec r\) as \(\vec r = x\hat i + y\hat j + z\hat k\). Further, the vector form of the equation of a line is \(\vec r = \vec a + λ\vec b\), can be converted to cartesian form of the equation of a line as \(\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}\). The vector form of equation of a plane \(\vec r.\vec n = d\), can be converted to cartesian form of the equation of the plane is ax + by + cz = d.

How To Represent A Point In Cartesian Form?

The point in a cartesian form is represented as (a, b, c), and each of it corresponds to the lengths along the x-axis, y-axis, and z-axis of the three-dimensional cartesian system

How To Write Equation Of A Line In Cartesian Form?

The cartesian form of equation of a line passing through the point \((x_1, y_1, z_1)\), and having the direction cosines a, b, c is \(\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} =\dfrac{z - z_1}{c}\). Also, another cartesian form of equation of a line passing through the points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), is \(\dfrac{x - x_1}{x_2 - x_1} =\dfrac{y - y_1}{y_2 - y_1} = \dfrac{z - z_1}{z_2 - z_1}\).

How To Write Equation Of A Plane In Cartesian Form?

The cartesian form of equation of a plane is ax + by + cz = d, where a, b, c are the direction ratios, and d is the distance of the plane from the origin.