F = ma & Vector Notation

Forces are vectors — they have both magnitude (in Newtons) and direction (usually an angle measured anti-clockwise from the horizontal).

In 2D, we use Cartesian coordinates ((x, y)) relative to an origin (O). Questions may say "Relative to the frame of reference Oxy" to indicate this.

![Force vector]()

Forces in 2D are written in the form (p\mathbf{i} + q\mathbf{j}) where (\mathbf{i}) and (\mathbf{j}) are unit vectors in the (x) and (y) directions.

• Column vectors (\begin{pmatrix} p \ q \end{pmatrix}) can also be used
• Both forms are equivalent for calculating resultants
• A force (\mathbf{F} = 3\mathbf{i} + 4\mathbf{j}) N has components 3 N horizontal and 4 N vertical

For a force (\mathbf{F} = x\mathbf{i} + y\mathbf{j}) N:

Magnitude: Use Pythagoras

[|\mathbf{F}| = \sqrt{x^2 + y^2}]

Direction: Use a diagram and trigonometry. Typically (\tan \theta = \frac{y}{x}) (treat (x) and (y) as positive for the ratio; adjust the angle depending on the quadrant).

When components are negative, still use (|x|) and (|y|) for the ratio (\frac{y}{x}), but adjust the angle depending on which quadrant the vector lies in.

Example: (\mathbf{F} = (-8\mathbf{i} - 6\mathbf{j})) N lies in the third quadrant. (|\mathbf{F}| = \sqrt{64+36} = 10) N. For direction from the positive horizontal: the angle is (180° + \arctan(6/8)).