Laws of Addition of Vectors

Two or more vectors can be added to give another vector, which is called the resultant of the vectors. The resultant would produce the same effect as that of the original vectors together.

(I)Geometrical method:

To find + , shift such that its initial point coincides with the terminal point of . Now, the vector whose initial point coincides with the initial point of , and terminal point coincides with the terminal point of represents ( + ) as shown in the above figure.

To find ( + ), shift such that its initial point coincides with the terminal point of . A vector whose initial point coincides with the initial point of and terminal point coincides with the terminal point of represents ( + ).

(II)Parallelogram law of addition of vectors:

If two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, their resultant will be represented in magnitude and direction by the diagonal of the parallelogram drawn from that point.

and are in the same direction i.e. they are parallel cos 0 ⁰ = 1

∴ | | = | | + | | & φ = 0 ⁰

(ii) θ = 180 ⁰ , and are in opposite direction i.e. they are antiparallel cos180 ⁰ = -1

∴ | | = | | ~ | | and is in the direction of the larger vector.

(iii) θ = 90 ^o , cos 90 ⁰ = 0

and are perpendicular to each other

∴ | | = ( | |2 + | |2)1/2 & φ= tan-1(Q/P)

Illustration 2. Two forces of 60N and 80N acting at an angle of 60 ⁰ with each other, pull an object. What single pull would replace the given forces?

∴ R ² = 60 ² + 80 ² + 2.60.80 cos 60 ⁰

= 3600 + 6400 + 4800 = 14800 ∴ R = 121.7N

Angle φ is given, tanφ =

Which gives, φ = 34.70

Illustration 3. The resultant of two vectors 3P and 2P is R. If the first vector is doubled, the resultant vector also becomes double. Find the angle between the vectors.

Solution: Let θ be the angle between vectors, then

= 13P ² + 12P ² cos θ …(1)

Also (2R) ² = (6P) ² + (2P) ² + 2(6P) (2P) cos θ

R ² = 10P ² + 6P ² cos θ …(2)

From (1) and (2)

cos θ = − 1/2

∴ θ = 120°

Illustration 4. The sum of magnitudes of two forces acting at a point is 18 and the magnitude of their resultant is 12. The resultant is at 90° with the force of smaller magnitude. What are the magnitude of individual forces.

Solution: Let and be two forces

Exercise 1:

i) Is it possible that the resultant of two equal forces is equal to one of the forces?

ii) If a vector has zero magnitude is it meaningful to call it a vector?

iii) Can three vectors, not in one plane, give a zero resultant? Can four vectors do?

Illustration 5. If the sum of two unit vectors and is also equal to a unit vector, find the magnitude of the vector .

Since force is directly proportional to acceleration,

Substituting | |, | | and a2/a1 = 2/1, we have,

= 2

Substituting θ = 1500 and solving the quadratic equation, we have

Illustration 7. The resultant of and is . If is doubled, is doubled; when is reversed, is again doubled, find P : Q : R.

Solution: Let θ be the angle between and . Then

R2 = | + |2 = P2 + Q2 + 2PQ cos θ … (i)

If is doubled, is doubled. That means, the magnitude of resultant of 2 and is 2R

(2R) ² = P ₂ + (2Q) ² + 2P(2Q) cos θ

This yields, 4R ² = P ₂ + 4Q ₂ + 4PQ cos θ

When is reversed, is doubled. Hence, the magnitude of resultant
and (- ) is 2R.

Then, (2R) ² = P ₂ + Q ₂ + 2PQ cos (!800 - θ).

This yields, 4R ₂ = P ₂ + Q ₂ – 2PQ cosθ … (iii)

eq. (i) – eq(iii) gives PQ cos θ = … (iv)

eq. (i) + eq (iii) gives P ₂ + Q ₂ = 5R2/2 …(v)

eq. (ii) + eq (iv) gives P ₂ + 4Q ₂ = 7R ₂

solving eq. (v) and eq(vi) we obtain Q = R and P = R

Hence P : Q : R =