Kinematics with Vectors | DP IB Maths: AI HL Revision Notes 2021

Kinematics using Vectors

How are vectors related to kinematics?

Kinematics is the use of mathematics to model motion in objects
If an object is moving in one dimension then its velocity, displacement and time are related using the formula s = vt

where s is displacement, v is velocity and t is the time taken

If an object is moving in more than one dimension then vectors are needed to represent its velocity and displacement

Whilst time is a scalar quantity, displacement and velocity are both vector quantities

Vectors are often used in questions in the context of forces, acceleration or velocity
The position of an object at a particular time can be modelled using a vector equation

How do I find the direction of a vector?

Vectors have opposite directions if they are the same size but opposite signs
The direction of a vector is what makes it more than just a scalar

E.g. two objects with velocities of 7 m/s and ‑7 m/s are travelling at the same speed but in opposite directions

Two vectors are parallel if and only if one is a scalar multiple of the other
For real-life contexts such as mechanics, direction can be calculated from a given vector using trigonometry

Given the i and j components a right-triangle can be created and the angle found using SOHCAHTOA

It is usually given as a bearing or as an angle calculated anticlockwise from the positive x-axis

How do I find the distance between two moving objects?

If two objects are moving with constant velocity in non-parallel directions the distance between them will change
The distance between them can be found by finding the magnitude of their position vectors at any point in time
The shortest distance between the two objects at a particular time can be found by finding the value of the time at which the magnitude is at its minimum value

Let the time when the objects are at the shortest distance be t
Find the distance, d, in terms of t by substituting into the equation for the magnitude of their position vectors
d² will be an expression in terms of t which can be differentiated and set to 0
Solving this will give the time at which the distance is at a minimum
Substitute this back into the expression for d to find the shortest distance

Exam Tip

Kinematics questions can have a lot of information in, read them carefully and pick out the parts that are essential to the question
Look out for where variables used are the same and/or different within vector equations, you will need to use different techniques to find these

Worked example

Two objects, A and B, are moving so that their position relative to a fixed point, O at time t, in minutes can be defined by the position vectors $r_{A} = (\binom{3}{- 1}) + t (\binom{- 2}{4})$ and $r_{B} = (\binom{2}{5}) + t (\binom{3}{- 1})$ .

The unit vectors i and j are a displacement of 1 metre due East and North of O respectively.

Find the coordinates of the initial position of the two objects.

3-9-1-ib-ai-hl-kin-vectors-we-soltuion-a

Find the shortest distance between the two objects and the time at which this will occur.

3-9-1-ib-ai-hl-kin-vectors-we-soltuion-b

Kinematics with Vectors (DP IB Maths: AI HL)

Revision Note