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

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What is the Vector Equation of a Line Passing Through Two Points?

Grade Level:

Class 12

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Definition

What is it?

The vector equation of a line passing through two points is like giving directions from one friend's house to another, using their exact locations. It describes every point on that straight path using a starting point and a direction vector. This equation helps us find any point on the line using a single parameter.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, living in different parts of your colony. Rohan's house is at point A and Priya's house is at point B. The vector equation of the line passing through their houses tells you how to draw a straight path from Rohan's house to Priya's house, and beyond. It helps locate any shop or landmark that falls exactly on this straight road.

Worked Example

Step-by-Step

Let's find the vector equation of a line passing through point A (1, 2, 3) and point B (4, 5, 6).

Step 1: Identify the position vectors of the two points. Let vector 'a' be the position vector of point A and vector 'b' be the position vector of point B.
a = i + 2j + 3k
b = 4i + 5j + 6k

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Step 2: Calculate the direction vector of the line. This vector goes from A to B. We find it by subtracting vector 'a' from vector 'b'.
Direction vector (b - a) = (4i + 5j + 6k) - (i + 2j + 3k)
Direction vector (b - a) = (4-1)i + (5-2)j + (6-3)k
Direction vector (b - a) = 3i + 3j + 3k

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Step 3: Write down the general form of the vector equation of a line passing through two points. It is given by r = a + lambda * (b - a), where 'r' is the position vector of any point on the line, 'a' is the position vector of the first point, 'b - a' is the direction vector, and 'lambda' is a scalar parameter.

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Step 4: Substitute the values we found into the general equation.
r = (i + 2j + 3k) + lambda * (3i + 3j + 3k)

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Answer: The vector equation of the line passing through points A and B is r = (i + 2j + 3k) + lambda * (3i + 3j + 3k).

Why It Matters

This concept helps engineers design flight paths for airplanes or routes for delivery drones, ensuring they fly in straight lines between two locations. In robotics, it guides robot arms to move precisely from one point to another. It's also vital for creating realistic 3D graphics in games or movies, where objects need to move along specific paths.

Common Mistakes

MISTAKE: Students sometimes subtract 'a' from 'b' incorrectly, mixing up the components or signs. | CORRECTION: Always subtract corresponding components (i from i, j from j, k from k) carefully, paying attention to positive and negative signs.

MISTAKE: Forgetting to include the 'lambda' (or 't') parameter in the final equation. | CORRECTION: The 'lambda' is crucial as it allows you to find any point on the line, not just the two given points. It must always be part of the equation.

MISTAKE: Using the direction vector (b - a) as the starting point 'a' in the equation. | CORRECTION: The 'a' in the equation r = a + lambda * (b - a) must be the position vector of one of the actual given points, not the direction vector.

Practice Questions

Try It Yourself

QUESTION: Find the vector equation of the line passing through P(2, 0, -1) and Q(3, 4, 1). | ANSWER: r = (2i - k) + lambda * (i + 4j + 2k)

QUESTION: A line passes through points A(5, 1, 2) and B(1, 3, 4). Write its vector equation. Then, find the coordinates of the point on this line when lambda = 0.5. | ANSWER: r = (5i + j + 2k) + lambda * (-4i + 2j + 2k). When lambda = 0.5, the point is (3i + 2j + 3k) or (3, 2, 3).

QUESTION: The vector equation of a line is r = (i + 2j) + lambda * (3i - j + 2k). If this line passes through two points P and Q, and P is (1, 2, 0), what could be the coordinates of Q? (Hint: Think about the direction vector.) | ANSWER: The direction vector is (3i - j + 2k). If P is (1, 2, 0), then Q could be P + (3i - j + 2k) = (1+3)i + (2-1)j + (0+2)k = (4i + j + 2k) or (4, 1, 2). (Other answers are possible by adding multiples of the direction vector).

MCQ

Quick Quiz

Which of the following represents the vector equation of a line passing through points with position vectors 'a' and 'b'?

r = a + b

r = lambda * (b - a)

r = a + lambda * (b - a)

r = b - a

The Correct Answer Is:

Option C is correct because it uses one point ('a') as a starting reference and adds a multiple (lambda) of the direction vector ('b - a') to reach any other point on the line. Options A, B, and D do not correctly represent a line through two points.

Real World Connection

In the Real World

When you use Google Maps or Ola/Uber to navigate, the app calculates the shortest, straight-line path between your current location and your destination. This path is essentially a line in 3D space. The algorithms behind these apps use vector equations to define these routes, helping your auto-rickshaw driver or delivery person find the most direct way to your home or a restaurant.

Key Vocabulary

Key Terms

VECTOR: A quantity having both magnitude and direction, like a force or velocity. | POSITION VECTOR: A vector that shows the position of a point in space relative to the origin. | DIRECTION VECTOR: A vector that indicates the direction of a line. | PARAMETER: A variable (like 'lambda') that can change, allowing the equation to represent many points on a line.

What's Next

What to Learn Next

Great job understanding this! Next, you can explore the Cartesian equation of a line passing through two points. It's another way to write the same line, using x, y, and z coordinates, and it builds directly on what you've learned here.