- Books Name
- Mathmatics Book Based on NCERT
- Publication
- KRISHNA PUBLICATIONS
- Course
- CBSE Class 12
- Subject
- Mathmatics
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 = (m–n)/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, r⃗ = (b⃗ +a⃗)/ 2. Hence, the position vector formula of mid-point R of PQ is,
OR= (b⃗ +a⃗ )/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
Also, the direction of p⃗ is the same (or opposite) to that of the line l, depending upon whether cosθ is positive or negative. The vector p⃗ is the projection vector and has magnitude |p⃗ |. It is also called the projection of 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,