Triangle Law of Vector Addition

Learn how two vectors combine using the triangle law,’with step-by-step visualization of their direction, magnitude, and resultant vector.

Triangle Law of Vector Addition

A vector can be added to another vector to find a resultant vector that represents their combined magnitude and direction. The triangle law of vector addition provides a simple graphical method for adding two vectors.

Triangle Law – states that if two vectors are represented by two sides of a triangle taken in order, then their resultant vector is represented by the third side of the triangle taken in the opposite direction.

First Vector – is drawn from the initial point to a second point, representing its magnitude and direction.

Second Vector – is drawn from the endpoint of the first vector, maintaining its given magnitude and direction.

Resultant Vector – is drawn from the starting point of the first vector to the endpoint of the second vector, representing the sum of the two vectors.

This law helps us understand the combined effect, magnitude, and direction of two vectors using a triangle.

Triangle law of vector addition Vector A goes from O to P, vector B goes from P to Q, and the resultant R goes from O to Q. Vector A Vector B Resultant R = A + B P O (start) Q (end)
Click “Replay steps” to see how the triangle is built.

Triangle Law of Vector Addition

Step 1 of 6
A B AB C BC AC

Triangle Law – Steps

In the diagram on the left, if AB displaces point A to point B,

and BC displaces point B to point C,

then their combined displacement is AC (displacement from point A to point C).

That is,

AB + BC = AC

Similarly,

AC + CB = AB   and   CA + AB = CB

This rule of vector addition is called the triangle law of vector addition.

Polygon Law of Vector Addition

Step 1 of 5
A B C D E AB BC CD DE AC CE AE

Polygon Law – Steps

Consider the pentagon ABCDE, in which the vectors AB, BC, CD and DE are taken in order along its sides.

Now, in pentagon ABCDE,

AB + BC + CD + DE

= AC + CD + DE   [From triangle law: AB + BC = AC]

= AC + CE   [From triangle law: CD + DE = CE]

= AE   [From triangle law: AC + CE = AE]

Therefore, AB + BC + CD + DE = AE.

This is the extended form of the triangle law of vector addition. It is called the polygon law of vector addition.

Example 1 of 2

Solve It