What is the Cartesian Equation of a Line? | Simple Explanation for Class 12

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What is the Cartesian Equation of a Line Passing Through a Point and Parallel to a Vector?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The Cartesian equation of a line passing through a point and parallel to a vector helps us describe a straight line in 3D space using coordinates (x, y, z). It shows how the coordinates of any point on the line are related to a known point on the line and the direction it's moving in.

Simple Example

Quick Example

Imagine you are at a specific spot on a cricket field (your 'point') and you want to hit the ball in a straight line towards the boundary (your 'vector'). The Cartesian equation tells you all the possible positions the ball can be in as it travels along that straight path towards the boundary. It gives the exact 'address' (x, y, z) of any point on that path.

Worked Example

Step-by-Step

Let's find the Cartesian equation of a line passing through point A(1, 2, 3) and parallel to the vector **b** = 2**i** + 3**j** + 4**k**.

Step 1: Identify the given point and the parallel vector. The given point is (x1, y1, z1) = (1, 2, 3). The direction ratios of the parallel vector are (a, b, c) = (2, 3, 4).
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Step 2: Recall the general form of the Cartesian equation of a line: (x - x1)/a = (y - y1)/b = (z - z1)/c.
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Step 3: Substitute the values of the point (x1, y1, z1) into the equation. This gives: (x - 1)/a = (y - 2)/b = (z - 3)/c.
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Step 4: Substitute the direction ratios (a, b, c) from the parallel vector into the equation. This gives: (x - 1)/2 = (y - 2)/3 = (z - 3)/4.
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Answer: The Cartesian equation of the line is (x - 1)/2 = (y - 2)/3 = (z - 3)/4.

Why It Matters

Understanding line equations is super important in fields like AI/ML for drawing decision boundaries, in Physics for tracking missile paths, and in Engineering for designing robotic arms. Scientists at ISRO use these equations to plot satellite trajectories, and doctors in Medicine use them for precise radiation therapy planning.

Common Mistakes

MISTAKE: Swapping the point coordinates with the direction ratios in the formula. For example, writing (x - a)/x1. | CORRECTION: Remember the formula is (x - x1)/a. The point (x1, y1, z1) always goes in the numerator, and the direction ratios (a, b, c) from the vector go in the denominator.

MISTAKE: Forgetting to subtract the point coordinates, and just writing x/a or x - x1/a. | CORRECTION: The numerator must always be (x - x1), (y - y1), and (z - z1). This shows the 'shift' from the origin to the given point.

MISTAKE: Mixing up the signs of the coordinates if the point or vector components are negative. For example, if the point is (-1, 2, 3), writing (x - 1). | CORRECTION: If the point is (-1, 2, 3), it should be (x - (-1)), which simplifies to (x + 1). Always be careful with negative signs.

Practice Questions

Try It Yourself

QUESTION: Find the Cartesian equation of a line passing through the point P(5, -2, 1) and parallel to the vector **v** = 3**i** + 0**j** - 2**k**. | ANSWER: (x - 5)/3 = (y + 2)/0 = (z - 1)/(-2) (Note: y+2/0 means the line is parallel to the YZ plane and x=5, z=1-2t)

QUESTION: A line passes through the origin (0, 0, 0) and is parallel to the vector **u** = **i** - 5**j** + 7**k**. Write its Cartesian equation. | ANSWER: x/1 = y/(-5) = z/7

QUESTION: If a line passes through point A(2, 4, 6) and is parallel to the line given by (x - 1)/3 = (y + 2)/4 = (z - 5)/(-1), find the Cartesian equation of the first line. | ANSWER: (x - 2)/3 = (y - 4)/4 = (z - 6)/(-1)

MCQ

Quick Quiz

Which of the following is the Cartesian equation of a line passing through point (1, -1, 2) and parallel to the vector 4**i** + 2**j** - 3**k**?

(x + 1)/4 = (y - 1)/2 = (z + 2)/(-3)

(x - 1)/4 = (y + 1)/2 = (z - 2)/(-3)

(x - 4)/1 = (y - 2)/(-1) = (z + 3)/2

(x - 1)/(-4) = (y + 1)/(-2) = (z - 2)/3

The Correct Answer Is:

The formula is (x - x1)/a = (y - y1)/b = (z - z1)/c. Here, (x1, y1, z1) = (1, -1, 2) and (a, b, c) = (4, 2, -3). Substituting these values gives (x - 1)/4 = (y - (-1))/2 = (z - 2)/(-3), which simplifies to (x - 1)/4 = (y + 1)/2 = (z - 2)/(-3).

Real World Connection

In the Real World

Imagine a drone delivering a package for a service like Dunzo or Zepto. To plan its flight path from the warehouse (a point) to your home (another point), while avoiding obstacles (following a specific direction), this equation is crucial. It helps define the exact straight line path the drone should follow in 3D space, ensuring efficient and safe delivery.

Key Vocabulary

Key Terms

CARTESIAN EQUATION: An equation that describes a geometric shape using coordinates (x, y, z) in a coordinate system. | POINT: A specific location in space, represented by its coordinates (x1, y1, z1). | VECTOR: A quantity with both magnitude (length) and direction, often used to describe movement or force. | DIRECTION RATIOS: The components of a vector that tell us its direction, denoted as (a, b, c). | 3D SPACE: A three-dimensional area where objects have length, width, and height.

What's Next

What to Learn Next

Great job understanding line equations! Next, you can explore the 'Vector Equation of a Line' and 'Equation of a Plane'. These build on what you've learned here and will help you describe more complex shapes and movements in 3D space, which is super useful for advanced math and science!