# Vectors in context

A moving object that has an initial position vector of ${\bf r}_0$ and is moving at a constant velocity of ${\bf v}$, has the following position vector at time t:

${\bf r}(t)={\bf r}_0+{\bf v}t$

A moving object that has an initial velocity vector of ${\bf v}_0$ and is moving with a constant acceleration of ${\bf a}$, has the following velocity vector at time t:

${\bf v}(t)={\bf v}_0+{\bf a}t$

Speed is found by taking the magnitude of the velocity vector.

Example 1Find the position vector, at time t = 3 minutes, of a plane that starts at position vector $3{\bf i}-4{\bf j}$ and moves at a constant velocity of $2{\bf i}+{\bf j}$ kilometres per minute where i and j are unit vectors in the x and y directions respectively.

Using the first formula above with ${\bf r}_0$ and ${\bf v}$ as given and t=3 gives ${\bf r}(3)=3{\bf i}-4{\bf j}+6{\bf i}+3{\bf j}=9{\bf i}-{\bf j}$.

Example 2A 12:00am, a ship has position vector $3{\bf i}-4{\bf j}$ where i and j are unit vectors in the x and y directions respectively. At 3:00pm, the ship has position vector $9{\bf i}-7{\bf j}$. Find the speed of the ship.

The ship moves by $6{\bf i}-3{\bf j}$ in three hours. This means that the shipping is moving $2{\bf i}-1{\bf j}$ every hour. This is a velocity vector and the corresponding speed is found from the magnitude of this vector:

Speed = $\sqrt{2^2+(-1)^2}=\sqrt{5}$ units per hour.