Vector Addition

Adding $5\ \mathrm{N}$ east to $5\ \mathrm{N}$ north does not give $10\ \mathrm{N}$ in a straight line. Because a vector has both magnitude and direction, vector addition has to track both, and for most introductory problems the reliable procedure is the component method: split each vector into horizontal and vertical parts, add matching parts, then rebuild the resultant.

When To Use This Method

The component method applies whenever you are combining quantities that are the same kind of vector. You can add displacement to displacement or force to force, but not force to velocity. If two vectors point the same way the resultant grows; opposite ways and it shrinks; at an angle and it points somewhere in between. Use components when the vectors meet at angles and you need a precise answer; the head-to-tail drawing is the quicker route when you only need intuition.

Step 1: Check The Quantity

Make sure the vectors describe the same physical quantity and use consistent units. This is also the moment to confirm you actually know each vector's direction clearly enough to add them.

Step 2: Choose A Method

The head-to-tail method is the visual route: draw the second vector starting from the head of the first; the resultant runs from the tail of the first to the head of the last. The component method is the calculation route: break each vector into horizontal and vertical parts. If the resultant has components $R_x$ and $R_y$, then

and the magnitude is

The direction comes from the component ratio, often $\tan \theta = R_y / R_x$ when the angle is measured from the positive $x$-axis.

Step 3: Add Matching Components

Add horizontal parts together and vertical parts together, keeping them separate, before you find the final magnitude and direction.

Step 4: Interpret The Result

The answer is a new vector, so report its direction as well as its size.

Full Walkthrough: Two Perpendicular Displacements

Suppose a student walks $3\ \mathrm{m}$ east, then $4\ \mathrm{m}$ north. Find the total displacement.

Check the quantity. Both are displacements in meters, so they add directly. Write components:

Add matching components:

Find magnitude and direction:

So the total displacement is $5\ \mathrm{m}$ at about $53^\circ$ north of east. This works cleanly because the two vectors are perpendicular, so the component picture is easy to read.

Where Each Step Goes Wrong

Adding magnitudes without checking direction. That only works when all vectors lie on the same line and point the same way. Otherwise, direction changes the result.

Mixing different physical quantities. You can add force to force or displacement to displacement, but not force to velocity.

Losing the direction in the final answer. A resultant is still a vector. Reporting only the magnitude is incomplete unless the problem explicitly asks for size alone.

Using a shortcut outside its condition. The $3$-$4$-$5$ triangle here works because the components are perpendicular, not because every pair of vectors forms a right triangle.

Practice The Same Steps

Redo the walkthrough with $6\ \mathrm{m}$ east and $8\ \mathrm{m}$ north, or make one vector point west instead of east, and predict the final direction before calculating. Then verify by computing the component sums first. For the first variation you should find a $10\ \mathrm{m}$ resultant at about $53^\circ$ north of east, the same direction as before but a larger magnitude.

Where Vector Addition Is Used In Physics

Vector addition appears whenever several directional effects combine into one result: total displacement after several moves, net force on an object, velocity relative to a moving medium, or electric and magnetic field contributions from different sources. In mechanics it is especially important for net force, since the vector sum of all forces on an object determines the overall effect on its motion.