Vector Algebra | Class 12 Mathematics Notes

Algebra of Vectors

The algebra of vectors involves operations such as addition, subtraction, and scalar multiplication. Vector addition is fundamental and can be performed geometrically or algebraically. Geometrically, vector addition is done using the triangle or parallelogram law. The triangle law states that if two vectors are represented by two sides of a triangle taken in order, then their sum is represented by the third side of the triangle taken in the opposite order. The parallelogram law states that if two vectors are represented by adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram passing through their common point. Algebraically, vectors are added by adding their corresponding components. Scalar multiplication involves multiplying a vector by a scalar, which changes the magnitude of the vector but not its direction unless the scalar is negative, which reverses the direction. Vector subtraction is defined as adding the negative of a vector. These operations obey properties such as commutativity and associativity, which are essential for vector manipulation.

📊 Diagram: Diagrams illustrating the triangle law and parallelogram law of vector addition, showing vectors as arrows and their resultant as the third side or diagonal respectively.

🧪 Activity: Activity: Using graphical methods to add two vectors and verify the result algebraically.

🔗 Connection: This section prepares the understanding required for 'Scalar (Dot) Product of Vectors' where multiplication of vectors resulting in scalars is introduced.