7 min read
Unit Vectors
A unit vector has length 1. In the direction of u (u not 0) it is u / |u|. For 6i+8j, |u|=10, so (35)i+(45)j. Check: (35)^2+(45)^2=1. The unit vector the other way is -u/|u|.
Choose Your Level
Unit Vectors
What is it?
A unit vector has length 1. In the direction of u (u not 0) it is u / |u|. For 6i+8j, |u|=10, so (35)i+(45)j. Check: (35)^2+(45)^2=1. The unit vector the other way is -u/|u|.
In simple words
- A unit vector has length 1.
- SAME DIRECTION: u = 6i+8j, |u|=10, unit = (610)i+(810)j = (35)i+(45)j.
- ZERO AND AXIS UNITS: No unit vector in the direction of 0.
- Worked-method reminder: check: (35)^2 + (45)2 = 925 + 1625 = 1
Important terms
UNIT VECTORS: A unit vector has length 1. In the direction of u (u not 0) it is u / |u|. For 6i+8j, |u|=10, so (35)i+(45)j. Check: (35)^2+(45)^2=1. The unit vector the other way is -u/|u|.
SAME DIRECTION: u = 6i+8j, |u|=10, unit = (610)i+(810)j = (35)i+(45)j. Do not skip simplifying 610.
ZERO AND AXIS UNITS: No unit vector in the direction of 0. i and j are already unit. A unit vector is not unique on a circle: many directions exist, each with its own unit.
What you should learn
- Explain unit vectors using the correct definitions, conditions, and notation.
- Apply u / |u|, length 1 correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- |6i+8j|=10
- 35 and 45
- Check squares sum to 1
SAME DIRECTION
u = 6i+8j, |u|=10, unit = (610)i+(810)j = (35)i+(45)j. Do not skip simplifying 610.
ZERO AND AXIS UNITS
No unit vector in the direction of 0. i and j are already unit. A unit vector is not unique on a circle: many directions exist, each with its own unit.
Recommended for Class 10
Full school-level explanation
A unit vector has length 1. In the direction of u (u not 0) it is u / |u|. For 6i+8j, |u|=10, so (35)i+(45)j. Check: (35)^2+(45)^2=1. The unit vector the other way is -u/|u|.
Divide by the length:
WHAT — UNIT VECTORS
A unit vector has length 1. In the direction of u (u not 0) it is u / |u|. For 6i+8j, |u|=10, so (35)i+(45)j. Check: (35)^2+(45)^2=1. The unit vector the other way is -u/|u|. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Unit Vectors matters because vectors describe both size and direction, so they are useful for displacement, velocity, force, and geometry. Components turn a directional problem into manageable algebra.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through SAME DIRECTION, then ZERO AND AXIS UNITS. Worked: unit vector in the direction of u = 6i + 8j. |u| = sqrt(36+64) = sqrt(100) = 10 u / |u| = (610)i + (810)j = (35)i + (45)j check: (35)^2 + (45)2 = 925 + 1625 = 1 Finish by checking signs, restrictions, units, and whether the answer matches the question.
SAME DIRECTION
u = 6i+8j, |u|=10, unit = (610)i+(810)j = (35)i+(45)j. Do not skip simplifying 610.
Remember
- divide each component
Examples
- |6i+8j|=10
- 35 and 45
- Check squares sum to 1
ZERO AND AXIS UNITS
No unit vector in the direction of 0. i and j are already unit. A unit vector is not unique on a circle: many directions exist, each with its own unit.
Remember
- 0 has no direction unit
Examples
- i and j are unit
- -i is unit the other way
- Not 6i+8j itself
Remember
- u / |u|
- length 1
- (35)i + (45)j
- u over |u|
- divide each component
- 0 has no direction unit
- u / |u|
- length 1
How the ideas connect
Worked: unit vector in the direction of u = 6i + 8j. |u| = sqrt(36+64) = sqrt(100) = 10 u / |u| = (610)i + (810)j = (35)i + (45)j check: (35)^2 + (45)2 = 925 + 1625 = 1
Deeper understanding
Go beyond memorising Unit Vectors. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: unit vector in the direction of u = 6i + 8j. |u| = sqrt(36+64) = sqrt(100) = 10 u / |u| = (610)i + (810)j = (35)i + (45)j check: (35)^2 + (45)2 = 925 + 1625 = 1
Common mistakes
Exam focus
Core Idea
SAME DIRECTION
ZERO AND AXIS UNITS
Worked Method
Exam focus
Quick revision
- u / |u|
- length 1
- (35)i + (45)j
- u over |u|
- UNIT VECTORS: A unit vector has length 1. In the direction of u (u not 0) it is u / |u|. For 6i+8j, |u|=10, so (3/
- SAME DIRECTION: u = 6i+8j, |u|=10, unit = (610)i+(810)j = (35)i+(45)j. Do not skip simplifying 610.
A unit vector is not 'the i and j of the problem'. It is any vector of length 1; the one 'in the direction of u' is specifically u/|u|.
For long answers on Unit Vectors: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- u / |u|
- length 1
- (35)i + (45)j
- u over |u|
Important formulas
- u / |u|
- length 1
- (35)i + (45)j
Common exam traps
- A unit vector is not 'the i and j of the problem'. It is any vector of length 1; the one 'in the direction of u' is specifically u/|u|.
- Not 6i+8j itself
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
A unit vector is not 'the i and j of the problem'. It is any vector of length 1; the one 'in the direction of u' is specifically u/|u|.
- Mistake
Not 6i+8j itself
- Mistake
Do not treat SAME DIRECTION and ZERO AND AXIS UNITS as the same step; each has a different purpose.
- Mistake
A unit vector is not 'the i and j of the problem'. It is any vector of length 1; the one 'in the direction of u' is specifically u/|u|.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
u / |u|
length 1
(35)i + (45)j
u over |u|
A unit vector has length 1. In the direction of u (u not 0) it is u / |u|. For 6i+8j, |u|=10, so (35)i+(45)j. Check: (35)^2+(45)2=1. The unit vector the ot
- Definition
- A unit vector has length 1. In the direction of u (u not 0) it is u / |u|. For 6i+8j, |u|=10, so (35)i+(45)j. Check: (35)^2+(45)2=1. The unit vector the ot
- Example
- |6i+8j|=10
- Critical warning
- A unit vector is not 'the i and j of the problem'. It is any vector of length 1; the one 'in the direction of u' is spec
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Ready to test yourself?
You've completed the learning section.
Exam Practice
Define
Q1. Define Unit Vectors.
Explain
Q2. Explain: SAME DIRECTION.
Example
Q3. Give one clear example of Unit Vectors and say why it fits.
Mistake
Q4. State one common mistake students make about Unit Vectors.
Exam note
Q5. Write a short exam note on Unit Vectors: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.