To add vectors, there are two common methods: graphically (geometrically) and algebraically (using components).
Graphical Method (Tip-to-Tail)
- Place the tail of the second vector at the tip of the first vector without changing their length or direction.
- Draw a vector from the tail of the first vector to the tip of the second vector. This new vector is the sum.
- This method visually combines vectors and works for any vector directions.
Algebraic Method (Using Components)
- If vectors are given in component form a=(ax,ay,az)\mathbf{a}=(a_x,a_y,a_z)a=(ax,ay,az) and b=(bx,by,bz)\mathbf{b}=(b_x,b_y,b_z)b=(bx,by,bz),
- Add corresponding components:
a+b=(ax+bx,ay+by,az+bz)\mathbf{a}+\mathbf{b}=(a_x+b_x,a_y+b_y,a_z+b_z)a+b=(ax+bx,ay+by,az+bz)
- This is often easier for calculations and yields the resultant vector components directly.
Both methods give the same result. The graphical method helps visualize vector addition, while the algebraic method is precise for calculations using coordinates or forces.
