Magnitude and unit vectors
The magnitude of a vector is its length. A unit vector has magnitude 1, so it records direction without keeping the original length.
Magnitude is Pythagoras on every component
For :
The vertical bars mean “magnitude”. The result is one non-negative scalar, not another vector.
Worked example
Length of a displacement in 3D
Answer
Common mistake
In 3D, include all three components. A negative component is squared, so its contribution to the length is positive.
Divide by the magnitude to make length 1
Keep the direction; change the length to 1
start
v
↓
find its length
|v| = √(x²+y²+z²)
↓
divide
v / |v|
The zero vector has no unit direction.
is read “v hat”. Divide every component by the same positive magnitude.
Worked example
A unit vector parallel to a given vector
Answer
- A positive multiple has the same direction; a negative multiple has the opposite direction.
- The zero vector has no direction, so it has no unit vector.
(Pure Mathematics 2 & 3 Coursebook, Ch. 9.1)
Build the requested direction before dividing
Your turnIndependent check [3 marks]
Find the unit vector from towards .
Show worked answer
Answer
Common mistake
“From P towards Q” requires , so calculate Q minus P. Reversing the subtraction gives the opposite unit vector.