Unit Vectors and Basis
Express vectors in the i, j, k basis and normalise a non-zero vector to preserve direction with magnitude one.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
i, j, k basis
Exam cue: What are the x-, y- and z-components?
Concept 2
component notation
Exam cue: What is the vector magnitude?
Concept 3
vector magnitude
Exam cue: Does the result have magnitude one?
Concept 4
unit vector
Concept 5
direction
Risk pitfalls and guardrails
Dividing by the magnitude squared
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Normalising the position point instead of the required displacement
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Dropping a negative component
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Basis names axes
i, j and k represent unit directions along x, y and z.
Components become coefficients
(a, b, c) is ai + bj + ck.
Unit means length one
A unit vector has magnitude exactly 1.
Divide by magnitude
For non-zero v, the unit direction is v/|v|.
Direction survives scaling
Positive normalisation changes length, not direction.
Check by magnitude
Square, add and root the final components to verify 1.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Find a unit vector in the direction of (3,4).
Find a unit vector in the direction of (−2,1,2).
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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