# 1.3.2 Vector Summation

3 kg + 4 kg always add up to be 7 kg. But 3 N + 4 N can be anything from 1 N to 7 N, depending on the direction of the two forces.

Triangle Method

To add or subtract two vectors, you can use the so-called triangle method, where the resultant vector is the side that closes the triangle.

Parallelogram Method

Alternatively, you can use the so-called parallelogram method, where the resultant vector is the diagonal of the parallelogram.

In General

After sketching the vector diagram, you can solve for the magnitude and direction of the resultant vector mathematically, typically using cosine rule or sine rule.

$\begin{array}{l}{{c}^{2}}={{a}^{2}}+{{b}^{2}}\\\tan \theta =\frac{a}{b}\end{array}$

Two Perpendicular Vectors

The calculation is even easier if the two vectors are perpendicular to each other. Since the resultant vector is the hypotenuse of a right angled triangle, all you need is Pythagoras and trigo ratios.

$\begin{array}{l}{{c}^{2}}={{a}^{2}}+{{b}^{2}}\\\tan \theta =\frac{a}{b}\end{array}$

Two Equal Magnitude Vectors

If the two vectors of the same magnitude, then the resultant vector is the diagonal of a rhombus, which again makes the calculation pretty simple.

$c=2a\cos \frac{\theta }{2}$

Video Explanation

Vector Summation

Concept Test

QQ0050