Cartesian Form of Vector - JEE Important Topics

The geometric entities in the cartesian plane can be represented on a three-dimensional plane, this representation is termed as cartesian form. It is represented in a three-dimensional form via X, Y, and Z axes. Along with this, we can also represent a point, a line, or a plane in a cartesian form. Let’s dig deeper into cartesian form and equations.

Cartesian Coordinates

Cartesian coordinates assist in discovering the precise spot of a factor in a 3D aircraft. The Cartesian coordinates of a factor are a couple of numbers which can be both in three dimensions or two dimensions. These numbers supply us the signed distances, which might be particularised, from the coordinate axis.

The Cartesian coordinates are in phrases of varieties of coordinate axes: 

• x-coordinates axis 

• y-coordinates axis

The foundation is the intersection of the x and y-axes. These coordinates are written down withinside the aircraft as (x, y). The x-coordinate is the signed distance from the foundation alongside the x-axis. This coordinate identifies the gap to the right or the left of the y-axis. When the gap is to the right side, x is positive, whilst it will be negative if the gap is to the left.

Quadrants

Cartesian Form of a Point

A point ‘L’ is represented as L(a, b, c) on a three-dimensional cartesian plane. Here a, b, and c are the co-ordinates of the point ‘L’ with respect to x-, y-, z-axes, respectively.

Cartesian Form of a Line

A line passing through the point $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ can be represented in a cartesian form as below.

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

Cartesian Form of a Plane

The cartesian form of a plane can be represented as ax + by + cz = d where a, b, and c are direction cosines that are normal to the plane and d is the distance from the origin to the plane.

Vector Line to Cartesian Form

The vector form can be easily converted into cartesian form by 2 simple methods.

(i) Using the arbitrary form of vector

$\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$

(ii) Using the product of unit vectors

Let us consider a arbitrary vector $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and an equation of the line that is passing through the points $\vec{a}$ and $\vec{b}$ is

$\vec{r}=\vec{a}+\lambda (\vec{b}- \vec{a})$

Suppose points $\vec{a}$ and $\vec{b}$ be (x1, y1, z1) and (x2, y2, z2). By substituting these in equation of a line we get,

$\vec{r}=\vec{a}+\lambda (\vec{b}- \vec{a})$

By taking $x_1,y_1, z_1$ co-ordinates to LHS we get,

$x\hat{i}+y\hat{j}+z\hat{k}={x_1\hat{i}+y_1\hat{j}+z_1\hat{k}}+\lambda((x_2 - x_1)\hat{i}+(y_2 - y_1)\hat{j}+(z_2 - z_1)\hat{k})$

$\Rightarrow (x - x_1)\hat{i}+(y - y_1)\hat{j}+(z - z_1)\hat{k}= \lambda((x_2 - x_1)\hat{i}+(y_2 - y_1)\hat{j}+(z_2 - z_1)\hat{k})$

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

Hence, the equation of line in cartesian form 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)}=\lambda$

Conclusion

Cartesian coordinates are the inspiration of analytic geometry and offer enlightening geometric interpretations for plenty of different branches of Mathematics, which include linear algebra, complicated analysis, differential geometry, multivariate calculus, institution idea, and more. This article helps us to understand what cartesian coordinates are and how we can represent points, lines, and planes in cartesian form. In addition to that, it also helps us to understand how to convert vector form to cartesian form.