Revision sheet

Vectors and components

• $\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle$: end minus start.
• Same components, same vector, wherever it starts.

Watch out: start minus end gives the opposite vector.

Magnitude and direction

• $|\langle a, b \rangle| = \sqrt{a^2 + b^2}$.
• $\tan\theta = \dfrac{b}{a}$; sketch the vector to pick the quadrant.
• Pointing left: add $180^\circ$ to the calculator's angle.

Watch out: the magnitude is not $a + b$.

Vector operations

• Add, subtract, scale: component by component.
• $\vec{u} + \vec{v}$: tip to tail.
• $|\vec{u} + \vec{v}|$: add first, then take the length.

Watch out: $|\vec{u} + \vec{v}| \ne |\vec{u}| + |\vec{v}|$ unless they point the same way.

From magnitude and direction

• $\langle r\cos\theta,\ r\sin\theta \rangle$.
• Unit vector: $\dfrac{\vec{v}}{|\vec{v}|}$, length 1.

Watch out: horizontal goes with cosine, vertical with sine.

The dot product

• $\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd$, a number.
• $\cos\theta = \dfrac{\vec{u} \cdot \vec{v}}{|\vec{u}|\,|\vec{v}|}$.
• $\vec{u} \cdot \vec{v} = 0$: perpendicular.

Watch out: multiply matching components, not crossed ones.

Vectors in context

1. Write each velocity or force in components.
2. Add them: the resultant.
3. Magnitude = speed or total force; $\tan\theta$ = direction.

Watch out: speeds in different directions do not simply add.