Section Formula

To begin with, take a look at the figure given below:

As shown above, P and Q are two points represented by position vectors OP and OQ, respectively, with respect to origin O. We can divide the line segment joining the points P and Q by a third point R in two ways:

  • Internally
  • Externally

If we want to find the position vector OR for the point R with respect to the origin O, then we should take both the cases one by one.

Case 1 – R Divides Segment PQ Internally

Take a look at Fig. 1 again. In this figure, if the point R divides such that,

where ‘m’ and ‘n’ are positive scalars, then we can say that R divides PQ internally in the ratio m:n. Now, from the triangles ORQ and OPR, we have

Therefore, replacing the values of and  in equation (1) above, we get

Hence, the position vector formula of the point R which divides PQ internally in the ratio m:n is,

 

Case 2 – R Divides Segment PQ Externally

Look at the figure given below:

In Fig. 2, point R divides the segment PQ externally in the ratio m:n. Hence, we can say that point Q divides PR internally in the ratio: (m – n) : n. Therefore,

PQQR = (mn)/n

Now, by using equation (2), we have

Note: If R is the Mid-Point of PQ

If R is the mid-point of PQ, then m = n. Therefore, from equation (2) above, we have

Therefore, r  = (b +a)/ 2. Hence, the position vector formula of mid-point R of PQ is,

OR= (b +a )/2 … (4)

Let’s look a solved example now:

Example 1

Consider two points P and Q with position vectors

Find the position vector formula of a point R which divides the line joining P and Q in the ratio 2:1, (i) internally, and (ii) externally.

Solution: Since point R divides PQ in the ratio 2:1. we have, m = 2 and n = 1

(i) R divides PQ internally

From equation (2), we have
 

Example:

Consider two points A and B with position vectors and

And

Find the position vector of a point C which divides the line joining A and B in the ratio 3 : 2, 

(i) internally

(ii) externally.

Solution:

(i) The position vector of the point C dividing the join of A and B internally in the ratio 3 : 2 is:

Expanding the terms in the numerator,

(ii) The position vector of the point C dividing the join of A and B externally in the ratio 3 : 2 is:

Expanding the terms in the numerator,

Projection of one vector on a line:

If a vector  makes an angle θ with a given directed line l, in the anticlockwise direction, then the projection of  on l is a vector p  with magnitude |cosθ.

Also, the direction of p  is the same (or opposite) to that of the line l, depending upon whether cosθ is positive or negative. The vector p  is the projection vector and has magnitude |p |. It is also called the projection of vector   on the directed line l.In each of the figures shown above, the projection vector of  along the line l is the vector: .

The vector projection of one vector over another vector is the length of the shadow of the given vector over another vector. It is obtained by multiplying the magnitude of the given vectors with the cosecant of the angle between the two vectors. The resultant of a vector projection formula is a scalar value.

Problem:-

Find the projection of the vector î – ĵ  on the vector î + ĵ.

Answer:-

Let a  = ( î – ĵ) and b  = ( î + ĵ)

Now, the projection of vector a  on b  is given by,

(1)/ (I b I) (a . b ) = (1)/ (√1+1) ({1.1 + (-1) (1)})

= (1/√2) (1-1) =0

Hence, the projection of vector a   on b  is  0.

 

Question: Represent graphically a displacement of 40 km, 30° east of north.

Solution:

The vector represents the displacement of 40 km, 30o east of north.

Question:  Find the unit vector in the direction of vector , where P and Q are the points

(1, 2, 3) and (4, 5, 6), respectively

Solution:

We know that,

QUESTION: Find a vector in the direction of vector which has magnitude 8 units.

Solution:

Firstly,