Vector Algebra: Graphical Method

To add two vectors using the parallelogram rule, position both vectors so their initial points coincide. Draw lines parallel to each vector from the endpoints of the other vector, forming a parallelogram. The diagonal drawn from the common initial point to the opposite corner is the resultant vector. This method works for any two vectors and produces the same result regardless of which vector you start with.

The tail-to-head method involves placing vectors sequentially so the tail of each new vector touches the head of the previous one. After arranging all vectors in this head-to-tail arrangement, draw the resultant by connecting the tail of the first vector to the head of the last vector. This generalizes vector addition for three or more vectors and produces the same resultant regardless of the order you arrange them.

To subtract vector B from vector A, first find the negative of vector B, which reverses its direction. Then add the negative vector to vector A using either the parallelogram rule or tail-to-head method. The resulting vector is the difference. The other diagonal of the parallelogram formed during addition represents this vector difference.

Multiplying a vector by a scalar produces a new vector quantity. If the scalar is positive, the resulting vector remains parallel to the original and scales in magnitude. If the scalar is negative, the resulting vector points in the opposite direction while maintaining the scaled magnitude. The direction depends entirely on the sign of the scalar.

The resultant vector's magnitude cannot equal a simple sum of component magnitudes because the diagonal of a parallelogram cannot be expressed as a simple sum of its side lengths. Geometric relationships determine the diagonal's length based on both the side lengths and the angle between them. This principle applies to both resultant and difference vectors formed through geometric construction.

The order of vector addition does not affect the resultant vector. Vector addition is both commutative and associative, meaning you can add vectors in any sequence and obtain the same final result. Whether you add vector A then B, or B then A, the magnitude and direction of the resultant remain identical.

When constructing a parallelogram from two vectors, one diagonal represents the resultant vector, while the other diagonal represents the vector difference. Both diagonals originate from the point where the vectors' initial points coincide. This single geometric construction simultaneously provides both the sum and difference of the two vectors using the cartesian coordinate system principles.