Worked example 1
Find a 3D unit vector
Find the unit vector in the direction of a = 2i − j + 2k.
- Calculate |a| = √(4 + 1 + 4) = 3
- Divide every component by 3.
- Keep the direction signs unchanged.
Answer: i − j + k
Higher Mathematics
Higher Maths 3D and unit vectors
Practise 3d and unit vectors for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Use i, j, k notation and find unit vectors in two or three dimensions.
Before you start
- Use 2D vector components.
- Calculate a magnitude with Pythagoras.
- Simplify fractions.
Explanation
Three-dimensional vectors use three components or i, j, k notation. Their magnitude extends Pythagoras to three perpendicular directions.
A unit vector has magnitude 1 and points in the same direction as the original non-zero vector.
Visual support
|a| = √(a₁² + a₂² + a₃²), unit vector =
Method and rules
- |ai + bj + ck| = √(a² + b² + c²)
- Unit vector in direction a = , for a not equal to 0
- ai + bj + ck corresponds to (a, b, c).
Worked examples
Worked example 2
Method check
Find the magnitude of i + 2j + 2k.
- Use √(1 + 4 + 4).
So: 3.
Watch out
- Dividing by the squared magnitude instead of the magnitude.
- Changing only one component.
- Trying to form a unit vector from the zero vector.
Exam reminder
For 3d and unit vectors, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
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