What Is the Resultant of Two or More Vectors?

The resultant vector is the single vector that has the same overall effect as two or more vectors acting together. It is found by calculating their vector sum, while preserving both magnitude and direction.

How the Resultant Is Found

For vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, the resultant $\mathbf{R}$ is

$$\mathbf{R}=\mathbf{a}+\mathbf{b}+\mathbf{c}.$$

Vectors are added component by component. For three-dimensional vectors,

$$(a_1,a_2,a_3)+(b_1,b_2,b_3)=(a_1+b_1,a_2+b_2,a_3+b_3).$$

Geometrically, place the tail of each new vector at the head of the previous vector. The resultant runs from the initial starting point to the final endpoint. This is called the head-to-tail method.

In an applied context, vectors might represent forces, velocities, or displacements. Their resultant describes the net force, net velocity, or overall displacement. Vector addition is commutative, so changing the order of the vectors does not change the resultant.

Worked Example

Suppose

$$\mathbf{a}=(3,-1),\qquad \mathbf{b}=(-2,4),\qquad \mathbf{c}=(1,5).$$

Then

$$\mathbf{R}=(3-2+1,-1+4+5)=(2,8).$$

Therefore, the combined movement is two units in the positive horizontal direction and eight units in the positive vertical direction. If the magnitude is required,

$$\lvert\mathbf{R}\rvert=\sqrt{2^2+8^2}=\sqrt{68}=2\sqrt{17}\approx 8.25.$$

A common misconception is to add the vectors' magnitudes. Magnitudes are scalars, so their sum generally does not equal the magnitude of the resultant. Direction must first be accounted for through signed components.

IB Exam Technique

This is HL-only content from AHL 3.10 that covers Vector definitions and may appear in any Math AI HL paper. Show the component addition before stating the resultant, preserve negative signs, and distinguish between the vector $\mathbf{R}$ and its magnitude $\lvert\mathbf{R}\rvert$. Give numerical magnitudes exactly or to three significant figures unless instructed otherwise.