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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.