Question

Prove the Section Formula for internal division in vectors.

Hint:

In internal division, multiply each endpoint vector by the opposite ratio and divide by the sum of ratios.

Answer 1

To Prove the Section Formula (Internal Division) in Vectors:

Let A and B be two points with position vectors

OA = a⃗ and OB = b⃗

Let P be a point dividing AB internally in the ratio m : n

That is,

AP : PB = m : n

We have to find the position vector of point P.

Step 1: Express AB in Vector Form

Vector AB = OB − OA

= b⃗ − a⃗

Step 2: Express AP in Terms of AB

Since P divides AB in the ratio m : n internally,

AP = (m / (m + n)) AB

Therefore,

AP = (m / (m + n)) (b⃗ − a⃗)

Step 3: Find OP

OP = OA + AP

= a⃗ + (m / (m + n))(b⃗ − a⃗)

= a⃗ + (m/(m+n))b⃗ − (m/(m+n))a⃗

= [ (m+n)/(m+n) ]a⃗ − (m/(m+n))a⃗ + (m/(m+n))b⃗

= (n/(m+n))a⃗ + (m/(m+n))b⃗

Therefore,

OP = ( n a⃗ + m b⃗ ) / (m + n)

Final Result (Section Formula – Internal Division):
If a point P divides the line segment joining A(a⃗) and B(b⃗) internally in the ratio m : n, then the position vector of P is

OP = ( n a⃗ + m b⃗ ) / (m + n)

Hence proved.