A Resultant as a Replacement Arrow
When several displacements occur in sequence, or several vectors act at one point, their vector sum can be represented by one arrow called the resultant. For forces applied at different points, the resultant force describes the net translational effect, but the turning effect must be checked separately through torque.
The graphical method finds the resultant using a scaled drawing. Arrow lengths are proportional to vector magnitudes, and arrow directions follow the given angles or compass directions.
In the graphical method, the accuracy depends on drawing scale, ruler work, and angle measurement.
Head to Tail Triangle Method
The triangle method is used to add two vectors. Draw the first vector, then place the tail of the second vector at the tip of the first. The resultant is drawn from the tail of the first vector to the tip of the second.
If someone walks east and then north, the resultant is not in one straight direction. It is the slanted arrow from the starting point to the final point.
Parallelogram Method for Two Vectors with One Tail
This method fits forces acting on one object from the same point, such as two people pulling a ring with ropes in different directions. In the drawing, both forces start from the same point, so the diagonal shows the combined pull.
Steps:
- Draw both vectors from the same tail.
- Copy the first vector at the tip of the second and copy the second vector at the tip of the first.
- Draw the diagonal from the shared tail to the far corner of the parallelogram.
Polygon Method for Many Vectors
The drawing order may be rearranged as long as each vector keeps its magnitude and direction. The resultant is drawn from the tail of the first chosen vector to the tip of the last vector.
If the polygon returns to the starting point, the vector sum is zero. For forces, zero net force means translational balance. Rotational balance additionally requires zero net torque.
Graphical Vector Subtraction
Vector subtraction does not need a new rule. Change subtraction into addition with a negative vector.
To draw , draw first. Then draw from the tip of . Vector has the same length as but points in the opposite direction.
The final result is still drawn from the tail of the first vector to the tip of the last vector. In this way, vector subtraction follows the same head-to-tail rule.