Mechanics - Combining and resolving vectors

Combining vectors

Vector quantities have direction as well as magnitude, so when they are combined the directions must be taken into account.

You can find the resultant of two vectors using scale drawings or by calculation. Take two forces, A and B, acting on a body. The two forces can be shown as two sides of a parallelogram. The diagonal of the parallelogram is the resultant.

It is often more convenient to draw a triangle. The two vector quantities are drawn 'tip-to-tail' and the resultant is the third side.

Combining Perpendicular Vectors

When two vectors are at right angles to each other, use Pythagorus' theorem and trigonometry to find the resultant.

First sketch the diagram. Then use a^2+b^2 = c^2 (c being the resultant quantity)

To find the angle, use SOH CAH TOA

Resolving Vectors

A single vector can be represented as the sum of two perpendicular vectors, known as its components. Vectors at right angles can then be treated independently, and can have their magnitude added to other parallel vectors if necessary.

The components of a vector are found by using that vector as the diagonal of a parallelogram of vectors, or as the hypotenuse of a right-angled triangle of vectors.

To find the horizontal quantity, use

cos(angle) = horizontal component/resultant

To find the vertical quantity, use

sin(angle) = vertical component/resultant

Combining more than two vectors

If there are more than two vectors, they can be combined by continuing the tip-to-tail procedure. The resultant is found by drawing from the tail of the first vector to the tip of the last.