Adding and subtracting vectors

5 minAS Level1.4.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can add and subtract coplanar vectors (vectors in the same plane) using a vector triangle, by scale drawing or by calculation.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$\alpha$	an angle

Tip to tail

To add vectors, you can't just add their sizes: 3 N + 4 N could be anything from 1 N to 7 N, depending on the directions. Instead:

Draw the first vector as an arrow, to scale and in the right direction.
Start the second vector at the tip of the first.
The resultant runs from the start of the first to the tip of the last.

The order doesn't matter: $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$ . The same method works for three or more vectors, chained tip to tail.

Calculating the resultant

The vectors are…	Use
along the same line	add with signs: $+5\ \text{N} + (-3\ \text{N}) = +2\ \text{N}$
at right angles	Pythagoras for the size, $\tan$ for the angle
at any other angle	an accurate scale drawing, or the cosine rule (and sine rule for the angle)

Worked example: A plane in a crosswind

A plane flies with a velocity of $120\ \text{m s}^{-1}$ due north relative to the air. A wind blows the air due east at $50\ \text{m s}^{-1}$ . Find the plane's resultant velocity.

Have a go first?

Quick check

CalculateQuestion 1 · Recall

Multiple choiceQuestion 2 · Apply

Answer the 2 questions above to continue.