Adding and subtracting vectors

G1.2

By the end of this lesson you should be able to

Solve vector problems by resolving vectors into components, adding or subtracting the components and recombining them to determine the resultant vector. G1.2

Key ideas

To add vectors on a diagram, join them head to tail; the resultant runs from the first tail to the last head.
To add by calculation: resolve every vector, add all the x-components, add all the y-components, then recombine.
Subtracting is adding the negative: \(\vec{a} - \vec{b} = \vec{a} + (-\vec{b})\). A change in a quantity is always final − initial.
Recombine with Pythagoras for the magnitude and \(\tan^{-1}\) for the direction. Check which quadrant the resultant is in.

Key formulas (QCAA formula book notation)

\[ R_x = \sum v_x \qquad R_y = \sum v_y \]

\[ R = \sqrt{R_x^2 + R_y^2} \qquad \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) \]

\[ \Delta\vec{v} = \vec{v}_f - \vec{v}_i \]

Worked example

A ship sails 40 km east, then 30 km at 60° north of east. Find its resultant displacement.

Common mistakes

Adding magnitudes directly (40 + 30 = 70 km) when the vectors point in different directions.
Using the calculator angle without checking the quadrant (add 180° when x is negative).
Doing initial − final for a change in velocity.

PhET: Vector Addition. Use the Explore 2D screen. Build two vectors, predict the resultant's components and size, then turn on the sum to check.

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