Add up to three vectors in component form and read off the resultant, its magnitude and direction, and the third vector needed to bring the total to zero.
| Vector | i | j | k |
|---|---|---|---|
| A | |||
| B | |||
| C optional |
Vectors add component by component. If A = a₁i + a₂j + a₃k and B = b₁i + b₂j + b₃k, then the resultant is (a₁+b₁)i + (a₂+b₂)j + (a₃+b₃)k. There is no shortcut with the magnitudes: adding lengths only works when the vectors point the same way.
|R| = √(R₁² + R₂² + R₃²)
A very common question gives you two vectors and asks what must be added so the resultant is zero. The answer is the negative of their sum:
C = −(A + B)
It has the same magnitude as the resultant of A and B, and points in exactly the opposite direction. This is the equilibrant — in a statics problem it is the force that holds the body in equilibrium.
The negative of their sum. Add the two vectors component by component, then reverse the sign of each component. The result has the same magnitude as their resultant but points the opposite way.
Add their corresponding components: the i components together, the j components together and the k components together. The magnitudes cannot simply be added unless the vectors are parallel.
Take the square root of the sum of the squares of its components. For a resultant of 2i plus 6j the magnitude is the square root of 4 plus 36, which is the square root of 40, about 6.32.
The vector that exactly cancels a resultant, equal in magnitude and opposite in direction. In a statics problem it is the force that keeps the body in equilibrium.