Vectors in mechanics

What you’ll learn

• Represent forces, positions and velocities in components.
• Use relative position and constant-velocity motion.

Understand the idea

A vector has magnitude and direction. In a plane, perpendicular unit vectors $\mathbf i$ and $\mathbf j$ let you work independently with horizontal and vertical components. The magnitude of a velocity vector is speed; its direction must be determined with the correct quadrant.

For constant velocity, add velocity multiplied by elapsed time to the initial position. Relative position is the difference of two position vectors. Two moving particles collide only if both components of their positions agree at the same time; crossing paths alone is insufficient.

Key formulae

$$|a\mathbf i+b\mathbf j|=\sqrt{a^2+b^2}$$

$$\mathbf r=\mathbf r_0+t\mathbf v$$

$$\mathbf r_{B/A}=\mathbf r_B-\mathbf r_A$$

$$\mathbf v_{B/A}=\mathbf v_B-\mathbf v_A$$

Worked example

A particle starts at $(2,-1)\,\mathrm m$ and moves with constant velocity $(3,2)\,\mathrm{m\,s^{-1}}$. Find its position after $4\,\mathrm s$.

1. The displacement in $4$ seconds is $4(3,2)=(12,8)$.
2. Add displacement to the initial position $(2,-1)$.
3. The coordinates are $(2+12,-1+8)=(14,7)$.

Answer: $(14,7)\,\mathrm m$.