Vector Operations

The parallelogram law states that two concurrent force vectors are added by placing their tails together and drawing lines parallel to each vector to form a parallelogram. The resultant vector is represented by the diagonal drawn from the common tail to the opposite corner. This method graphically determines both the magnitude and direction of the combined force effect.

The triangle rule arranges two vectors head-to-tail as consecutive sides of a triangle, with the third side representing the resultant vector. Unlike the parallelogram law, which places tails together, the triangle rule connects vectors sequentially. Both methods yield identical resultant magnitude and direction, offering alternative geometric approaches to vector addition.

When a vector is multiplied by a negative scalar, the magnitude changes by the scalar's absolute value, and the direction reverses completely. For example, multiplying a velocity vector by negative mass reverses the resulting momentum vector's direction while scaling its magnitude. Positive scalars preserve direction while only changing magnitude.

Vector subtraction is performed by adding the reverse, or negative, of one vector to another. This means reversing the direction of the vector being subtracted, then applying standard vector addition rules such as the parallelogram or triangle law. The resulting difference vector represents the geometric outcome of this operation.

Collinear vectors share the same line of action, meaning they act along the same direction or opposite directions. When collinear vectors are added, the parallelogram law simplifies to algebraic or scalar addition, where magnitudes are combined directly. This special case eliminates the need for geometric construction methods.

Scalar multiplication changes a vector's magnitude by the scalar value while preserving or reversing its direction. A positive scalar maintains the original direction and scales magnitude proportionally. A negative scalar reverses direction and scales magnitude, producing a new vector with altered properties but same line of action.

Momentum is calculated by multiplying mass, a scalar quantity, by velocity, a vector quantity. The resulting momentum vector has the same direction as velocity but magnitude scaled by the mass value. This demonstrates how scalar multiplication produces physically meaningful vectors in engineering applications.