What is Section Formula for External Division? | Simple Explanation for Class 12

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What is the Section Formula for External Division in Vectors?

Grade Level:

Class 12

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Definition

What is it?

The Section Formula for External Division in Vectors helps us find the position vector of a point that divides a line segment externally in a given ratio. Imagine a point lying outside the line segment, extending it, and dividing it in a specific proportion. This formula tells us exactly where that external point is located.

Simple Example

Quick Example

Imagine two friends, Rohan and Priya, standing at different spots. Rohan is at point A and Priya is at point B. A third friend, Sameer, is standing outside the line connecting Rohan and Priya, but in such a way that his distance from Rohan is twice his distance from Priya (2:1 ratio). The Section Formula for External Division helps us find Sameer's exact location.

Worked Example

Step-by-Step

Let point A be (1, 2, 3) and point B be (4, 5, 6). We want to find the position vector of point R that divides the line segment AB externally in the ratio 2:1.

Step 1: Identify the position vectors. Let vector 'a' be the position vector of A, and vector 'b' be the position vector of B. So, a = i + 2j + 3k and b = 4i + 5j + 6k.
---Step 2: Identify the ratio m:n. Here, m = 2 and n = 1.
---Step 3: Recall the external division formula: R = (m*b - n*a) / (m - n).
---Step 4: Substitute the values into the formula: R = (2 * (4i + 5j + 6k) - 1 * (i + 2j + 3k)) / (2 - 1).
---Step 5: Simplify the numerator: R = (8i + 10j + 12k - i - 2j - 3k) / 1.
---Step 6: Combine like terms: R = (8i - i) + (10j - 2j) + (12k - 3k).
---Step 7: Final calculation: R = 7i + 8j + 9k.

Answer: The position vector of point R is 7i + 8j + 9k.

Why It Matters

This formula is crucial in fields like Engineering and Physics to calculate the exact location of forces or centers of mass. In AI/ML, it helps in understanding data point distributions. Even in game development, it's used to position objects or characters precisely in a virtual world.

Common Mistakes

MISTAKE: Using the internal division formula instead of external. | CORRECTION: Remember, for external division, there's a minus sign in both the numerator and denominator: (m*b - n*a) / (m - n).

MISTAKE: Swapping 'm' and 'n' or 'a' and 'b' in the formula. | CORRECTION: The ratio is m:n, meaning 'm' is associated with the vector 'b' (the second point) and 'n' with vector 'a' (the first point).

MISTAKE: Forgetting to subtract 'n*a' from 'm*b' and performing addition instead. | CORRECTION: Always ensure you subtract 'n*a' from 'm*b' in the numerator for external division.

Practice Questions

Try It Yourself

QUESTION: Find the position vector of a point R that divides the line segment joining P(1,0,2) and Q(3,4,0) externally in the ratio 3:1. | ANSWER: R = 4i + 6j - k

QUESTION: Given points A with position vector 2i - j and B with position vector 5i + 2j. Find the position vector of point C that divides AB externally in the ratio 2:3. | ANSWER: C = -i - 5j

QUESTION: If point R with position vector 7i + 8j + 9k divides the line segment joining A(1,2,3) and B(x,y,z) externally in the ratio 2:1, find the position vector of B. | ANSWER: B = 4i + 5j + 6k

MCQ

Quick Quiz

Which of the following is the correct formula for the position vector 'r' of a point dividing the line segment joining vectors 'a' and 'b' externally in the ratio m:n?

(m*a + n*b) / (m + n)

(m*b - n*a) / (m - n)

(n*a - m*b) / (n - m)

(m*b + n*a) / (m + n)

The Correct Answer Is:

Option B is the correct formula for external division. Options A and D are for internal division, and option C incorrectly swaps the terms and signs for external division.

Real World Connection

In the Real World

Think about how delivery apps like Zomato or Swiggy calculate optimal routes. If a delivery person needs to pick up an order from point A and then go to point B, but there's a temporary 'hotspot' or a specific point they need to avoid (an external point) to optimize fuel or time, this formula can be used to model and plan such routes efficiently. It helps in understanding positions relative to other positions, even if they are 'outside' the direct path.

Key Vocabulary

Key Terms

POSITION VECTOR: A vector that shows the position of a point from the origin. | EXTERNAL DIVISION: A point dividing a line segment by lying outside the segment. | RATIO (m:n): The proportion in which the point divides the line segment. | LINE SEGMENT: A part of a line that is bounded by two distinct end points.

What's Next

What to Learn Next

Now that you've mastered external division, you're ready to explore the Section Formula for Internal Division. Understanding both will give you a complete picture of how points divide line segments, which is super useful for more complex geometry problems!