If two vectors are acting on a body at the same time and are represented in both magnitude and direction by two sides of a triangle in sequence, then their resultant vector (in terms of both magnitude and direction) is represented by the third side of the triangle taken in the reverse order.
This method is specifically applicable for vector addition.
Steps for adding two vector representing same physical quantity by triangle law:
(i) (ii) (iii)
Q1. A boy travels 10 m due to North and then 7m due to East. Find the displacement and direction of body.
Sol: Let the boy start moving from point O as shown in the figure.
where, OA = 10 m, due North
AB = 7 m, due East
According to triangle law of vector addition, OB is the resultant displacement.
The magnitude of the resultant displacement,
Since, the resultant displacement makes an angle θ with the North direction. Then,
This law states that if two vectors are positioned as adjacent sides of a parallelogram with their tails connected, the sum of these two vectors will be represented by the diagonal of the parallelogram, starting from the same point as the two vectors.
Consider the vectors P and Q in the figure below. To find their sum:
Note: Angle between 2 vectors is the angle between their positive directions.
Suppose angle between these two vectors is θ, and
(AD)^{2} = (AE)^{2} +(DE)^{2}
= (AB + BE)^{2} + (DE)^{2}
= (a +b cosθ)^{2} + (b sinθ)^{2 }
= a^{2} + b^{2} cos^{2}θ + 2ab cosθ + b^{2} sin^{2}θ
= a^{2} + b^{2} + 2ab cosθ
Thus, AD =
or
Angle α with vector a is
tan α = =
Important points :
Q2. A body is simultaneously given two velocities of 30 m/s due East and 40 m/s due North, respectively. Find the resultant velocity.
Sol: Let the body be starting from point O as shown.
This law is used for adding more than two vectors. This is extension of triangle law of addition. We keep on arranging vectors such that tail of next vector lies on head of former.
When we connect the tail of first vector to head of last we get resultant of all the vectors.
Properties of Addition of Vectors:
Properties of Subtraction of Vectors:
Q3. Two vectors of 10 units & 5 units make an angle of 120° with each other. Find the magnitude & angle of resultant with vector of 10 unit magnitude.
Sol: =
⇒ α = 30°
[Here shows what is angle between both vectors = 120° and not 60°]
Note: or can also be found by making triangles as shown in the figure. (a) and (b)
Or
Q4. Two vectors of equal magnitude 2 are at an angle of 60° to each other. Find the magnitude of their sum & difference.
Sol:
Q5. Find and in the diagram shown in figure. Given A = 4 units and B = 3 units.
Sol: Addition :
R =
= = units
tanα = = = 0.472
a = tan^{1}(0.472) = 25.3°
Thus, resultant of and is units at angle 25.3° from in the direction shown in figure.
Subtraction : S =
= =
and tanθ =
= = 1.04
∴ α = tan^{1} (1.04) = 46.1°
Thus, is units at 46.1° from in the direction shown in figure.
261 videos249 docs232 tests

1. What is the Triangle Law of Vector Addition? 
2. How does the Parallelogram Law of Vector Addition work? 
3. What is the Polygon Law of Vector Addition? 
4. How can vectors be subtracted graphically? 
5. What are the applications of vector addition and subtraction in real life? 

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