The triangle law of vector addition says that when two vectors are represented as two sides of a triangle with the same order of magnitude and direction, then the magnitude and direction of the resultant vector is represented by the third side of the triangle taken in reverse order.
Vector Addition is the process of combining two or more vectors to determine a resultant vector, which represents the overall effect of those vectors. Vectors are quantities that have both magnitude and direction, such as force, velocity, or displacement. Unlike scalar quantities, which can be simply added arithmetically, vectors require both magnitude and direction to be considered in their addition.
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There are two primary methods of vector addition: the Triangle Law and the Parallelogram Law. In both cases, the individual vectors are placed tail to head, and the resultant vector is drawn from the starting point to the endpoint.
Vectors are written/represented with an alphabet and an arrow over them and are represented as a combination of direction and magnitude. The addition of two or more vectors is referred to as vector addition. When we add vectors, we use the addition operation to add two or more vectors to obtain a new vector that equals the sum of the two or more vectors. Vector addition can be used to combine two vectors, a and b, and thus the resultant vector can be expressed as:
R=a+ b here bold is used to show that R, a and b are vectors
There are different laws of vector addition and these are:
The triangle law of vector addition states that if two vectors are represented by the sides of a triangle taken in order of magnitude and direction, then the resultant sum of the vectors is given by the triangle's third side in reverse order of magnitude and direction.
Mathematically, if you have two vectors A and B, and you place the tail of vector B at the head of vector A, then the resultant vector R is the vector drawn from the tail of A to the head of B. The direction and magnitude of this resultant vector represent the combined effect of both vectors.
The Parallelogram Law of Vector Addition is another fundamental method for determining the resultant of two vectors. This law states that if two vectors are represented as adjacent sides of a parallelogram, then the diagonal of the parallelogram that starts from the common point of the vectors represents their resultant.
Here's how it works:
The Triangle Law of forces is applicable when there are three forces acting on a body in equilibrium. The two forces are then represented as two sides of a triangle in the same order, with their magnitude scaled to a suitable scale, and the resultant in the opposite order is the third side or closing side of the triangle.
The Triangle Law of Forces can also be used to calculate the resultant of two forces acting at a point.
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Consider two vectors P and Q , represented in both magnitude and direction by the sides OA and AB of a triangle OAB, respectively. Let R be the product/resultant of the triangle law of vector additions. The resultant of P and Q is therefore represented by side OB according to the triangle law of addition or triangle law of vector addition.
We have,
Expand A to C and draw perpendicular BC .
From triangle OCB we have,
In the triangle ACB,
Also,
Resultant magnitude:
Substitute the values for
The above equation gives us the magnitude of the resultant.
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The direction of the magnitude of the resultant vector is given by;
From triangle OBC ,
Since,
The above equation gives us the direction of the resultant vector.
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Example: Two vectors A and B of magnitude 5 units and 7 units respectively make an angle of 60o. Determine the magnitude of the resultant vector.
Solution: By following the triangle law of vector addition, the resultant vector is given by:
The magnitude of
Here is a list of some points to keep in mind while studying vector addition:
Vectors are depicted with an arrow and are represented as a combination of direction and magnitude.
If we know the components of a vector, we can calculate the direction of the resultant vector.
The well-known triangle law of forces can be used to add vectors, and this method is also known as the head-to-tail method.
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NCERT Physics Notes:
The sum of two vectors can be found mathematically using the triangular law of vector addition. Mathematical physics includes vector addition and subtraction as essential concepts. A vector is an entity with both magnitude and direction, sometimes known as a quantity. It is a geometric addition of the vectors.
The Triangle Law of Vector Addition states that when two vectors are represented by two sides of a triangle in order of magnitude and direction, the magnitude and direction of the resultant vector are represented by the third side of the triangle.
Vector addition meets two important properties
1. In commutative law, the order of addition is irrelevant, so A+B is equal to B+A.
2. The Associative law states that the sum of three vectors is independent of which pair of vectors is added first, i.e. (A+B)+C=A+(B+C).
According to the Parallelogram law of vector addition, if any two vectors a and b represent two sides of a parallelogram in magnitude and direction, then their sum a + b equals the diagonal of the parallelogram across their common point in magnitude and direction.
The commutative property of vector addition asserts that for any two vectors a and b,
a + b = b + a.
According to the Associative Property of Vector addition, for any three vectors a, b, and c,
(a + b ) + c = a + (b + c )
The magnitude of a vector denotes the length of the vector. The magnitude of the vector a is denoted as |a|.
Polygon law of vector addition states that if the number of vectors can be represented in magnitude as well as direction by side of a polygon taken in the same order, then their resultant is represented in magnitude along with direction by the closing side of the polygon taken in the opposite order
The magnitude of the resultant vector can be calculated using the formula
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