What is Section Formula (Internal Division) in Vectors? | Class 12

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What is the Section Formula for Internal Division in Vectors?

Grade Level:

Class 12

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Definition

What is it?

The Section Formula for Internal Division in Vectors helps us find the position vector of a point that divides a line segment joining two other points internally in a given ratio. Imagine you have two friends standing at different spots, and you want to find the exact spot a third friend needs to stand *between* them to divide their distance in a specific way.

Simple Example

Quick Example

Suppose your school is at point A and your home is at point B. Your friend's house is at point C, exactly on the road between your school and home. If your friend's house divides the road from school to home in the ratio 1:2 (meaning it's closer to school), the Section Formula helps find the exact location of your friend's house.

Worked Example

Step-by-Step

Let's find the position vector of a point R that divides the line segment joining points P (with position vector 2i + 3j + k) and Q (with position vector 5i + 6j + 4k) internally in the ratio 1:2.

Step 1: Identify the position vectors of the two points and the ratio. Here, vector P = 2i + 3j + k, vector Q = 5i + 6j + 4k. The ratio m:n is 1:2, so m=1 and n=2.
---Step 2: Recall the Section Formula for internal division: R = (n * P + m * Q) / (m + n).
---Step 3: Substitute the values into the formula: R = (2 * (2i + 3j + k) + 1 * (5i + 6j + 4k)) / (1 + 2).
---Step 4: Distribute the numbers: R = (4i + 6j + 2k + 5i + 6j + 4k) / 3.
---Step 5: Combine like terms (i with i, j with j, k with k): R = ( (4+5)i + (6+6)j + (2+4)k ) / 3.
---Step 6: Simplify: R = (9i + 12j + 6k) / 3.
---Step 7: Divide each component by 3: R = 3i + 4j + 2k.
---Answer: The position vector of point R is 3i + 4j + 2k.

Why It Matters

This formula is super useful in fields like engineering and robotics to calculate exact positions. For example, in AI/ML, it helps robots know where to move their 'hands' or 'legs' to pick up an object precisely. Engineers use it to design structures like bridges, ensuring parts are connected at the right spots.

Common Mistakes

MISTAKE: Swapping m and n in the formula, applying n to the second vector and m to the first. | CORRECTION: Remember the formula is (n*vector1 + m*vector2) / (m+n). The 'n' multiplies the first point's vector, and 'm' multiplies the second point's vector.

MISTAKE: Forgetting to add m+n in the denominator. | CORRECTION: The sum of the ratios (m+n) must always be in the denominator to correctly 'average' the vectors according to the ratio.

MISTAKE: Making calculation errors when distributing the ratio numbers to the vector components (i, j, k). | CORRECTION: Distribute carefully. Multiply 'n' with *all* components of the first vector, and 'm' with *all* components of the second vector, then combine.

Practice Questions

Try It Yourself

QUESTION: Find the position vector of a point R that divides the line segment joining points A (with position vector i + 2j) and B (with position vector 4i + 5j) internally in the ratio 1:1. | ANSWER: (2.5i + 3.5j)

QUESTION: Point C divides the line segment joining P (3i - j + 2k) and Q (i + 5j - 4k) internally in the ratio 2:3. What is the position vector of C? | ANSWER: (2.2i + 1.4j + 0.4k)

QUESTION: A drone is flying from point X (position vector 2i + 2j + 2k) to point Y (position vector 8i + 8j + 8k). A sensor needs to be placed at a point S on its path such that S divides the path in the ratio 3:1 (closer to Y). Find the position vector of S. | ANSWER: (6.5i + 6.5j + 6.5k)

MCQ

Quick Quiz

If point P divides the line segment joining vectors A and B internally in the ratio m:n, what is the correct formula for the position vector of P?

(m*A + n*B) / (m+n)

(n*A + m*B) / (m+n)

(m*A - n*B) / (m-n)

(n*A - m*B) / (m-n)

The Correct Answer Is:

The correct formula for internal division is (n*A + m*B) / (m+n). The ratio 'n' is multiplied with the first vector (A) and 'm' with the second vector (B). Options C and D are for external division or have incorrect signs.

Real World Connection

In the Real World

Think about how Zomato or Swiggy deliver food! They use complex algorithms that often involve vector math. If a delivery executive needs to pick up food from restaurant A and deliver to customer B, the app might calculate an intermediate stop C (like a fuel station) that divides the path in a certain ratio, perhaps to optimize route or time. This formula helps locate such intermediate points precisely.

Key Vocabulary

Key Terms

POSITION VECTOR: A vector that shows the position of a point from the origin. | INTERNAL DIVISION: When a point lies *between* two other points on a line segment. | RATIO: A comparison of two numbers, like m:n. | LINE SEGMENT: A part of a line that has two endpoints.

What's Next

What to Learn Next

Now that you understand internal division, you can explore the Section Formula for External Division in Vectors. This is similar but deals with a point dividing a line segment from *outside* the segment, which is also very useful in advanced geometry and physics problems!