What Is the Cartesian Form of a Complex Number?
Understanding Cartesian Form will help you easily represent points, lines, and planes on coordinate geometry questions in school and exams. This is a fundamental skill for board and competitive maths, making 2D and 3D geometry problems much simpler to solve with clear formulas and steps.
Formula Used in Cartesian Form
The standard formulas are:
For a point: \( (x, y, z) \)
For a line: \( \dfrac{x-x_1}{a} = \dfrac{y-y_1}{b} = \dfrac{z-z_1}{c} \)
For a plane: \( ax + by + cz = d \)
Here’s a helpful table to understand Cartesian Form more clearly:
Cartesian Form Table
| Entity | Cartesian Form | Dimension |
|---|---|---|
| Point | (x, y, z) | 1D / 2D / 3D |
| Line | \( \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \) | 3D |
| Plane | \( ax + by + cz = d \) | 3D |
| Circle (2D) | \( (x-h)^2 + (y-k)^2 = r^2 \) | 2D |
This table shows how the pattern of Cartesian Form is applied to different geometric elements for coordinate geometry in both two and three dimensions.
Worked Example – Solving a Problem
Let’s write the equation of a line passing through points (2, 3, −1) and (5, 6, 7) in Cartesian Form.
1. Start by identifying the two points: \((x_1, y_1, z_1) = (2, 3, -1)\) and \((x_2, y_2, z_2) = (5, 6, 7)\).2. Apply the formula: \( \frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1} \).
3. Substitute the values:
4. Simplify each denominator:
5. Final answer:
For more solved examples, visit our page on General Equation of a Line and Equation of a Plane.
Practice Problems
- Write the equation of a plane passing through (1, 2, 3) with direction ratios 2, −1, 4 in Cartesian Form.
- Convert the vector \( \vec{r} = 5\hat{i} + 7\hat{j} - 2\hat{k} \) to Cartesian coordinates.
- Find the Cartesian Form of a line that passes through the points (0, 1, 1) and (2, 4, −3).
- Express the equation \( x^2 + y^2 = 16 \) as a circle in Cartesian Form.
You can boost your practice by reviewing concepts on the Coordinate Geometry page or testing your understanding on Cartesian Plane examples.
Common Mistakes to Avoid
- Mixing up Cartesian Form with vector or polar forms.
- Using the wrong coordinate for the direction ratios or differences between points.
- Forgetting if the problem is 2D or 3D—always check if you need (x, y) or (x, y, z).
Real-World Applications
The concept of Cartesian Form is crucial in fields like mapping, navigation, engineering design, and computer graphics. With Cartesian coordinates, engineers and programmers model objects with accuracy. Vedantu guides students to use these forms for visualising maths in the real world as well as for scoring better in exams.
Related Topics & Internal Links
To deepen your knowledge of Cartesian Form, review these important concepts:
- Cartesian Coordinates – Learn how every point is located in space.
- Two-Dimensional Coordinate Geometry – Apply Cartesian Form in 2D.
- Coordinate Grid – Visualise how the grid helps in plotting points.
- Intercepts of a Line – See how Cartesian Form assists in line equations.
- Line Equation – Point Slope – Connect coordinate form and slopes.
We explored the idea of Cartesian Form, how to use its equations, solve problems, and see practical uses. Practice these ideas regularly with Vedantu and check related concepts to become confident in all types of coordinate geometry questions!
FAQs on How to Convert to Cartesian Form: Step-by-Step Guide
1. How do you convert a complex number to Cartesian form?
To convert a complex number to Cartesian form, express it as z = x + iy, where x is the real part and y is the imaginary part. If you have the polar form (r, θ), use the formulas: x = r cosθ and y = r sinθ, then write as z = x + iy.
2. What is a Cartesian example?
A typical example of a number in Cartesian form is z = 3 + 4i, where 3 represents the real part and 4 is the imaginary part.
3. What is the Cartesian form of Z?
The Cartesian form of Z is written as Z = x + iy, where x is the real part and y is the imaginary part of the complex number.
4. What is the Cartesian form of a number?
The Cartesian form of a complex number is z = x + iy. Here, x and y are both real numbers, representing the horizontal and vertical coordinates on the complex plane.
5. What is the Cartesian formula?
The standard Cartesian formula represents mathematical objects using x and y coordinates. For example, a line is given by y = mx + c, while a point is given as (x, y) in the Cartesian plane.
6. What is the Cartesian form of a vector?
The Cartesian form of a vector is written as →A = xi + yj + zk in three dimensions, where x, y, and z are the scalar components along the respective axes.
7. How do you convert Cartesian form to polar form?
To convert a number from Cartesian form (x + iy) to polar form (r, θ), use these steps:
1. r = √(x² + y²) (magnitude)
2. θ = tan-1(y/x) (argument).
Then write as r (cosθ + i sinθ).
8. What is the Cartesian equation of a circle?
The Cartesian equation of a circle with centre at (h, k) and radius r is (x − h)² + (y − k)² = r².
9. What is the Cartesian form of an ellipse?
The Cartesian equation of an ellipse with center (h, k) is ((x − h)² / a²) + ((y − k)² / b²) = 1, where a and b are the semi-axes.
10. What is the difference between Cartesian form and polar form?
In Cartesian form, numbers or points are expressed as (x, y) or x + iy. In polar form, they are given as (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis.
11. How do you convert a vector from polar to Cartesian form?
To convert a vector from polar form (r, θ) to Cartesian form, use:
1. x = r cosθ
2. y = r sinθ
Then write the vector as xi + yj.
12. What is the Cartesian form of a plane in 3D geometry?
The Cartesian equation of a plane in 3D is given by ax + by + cz + d = 0, where a, b, c, and d are constants.
