MathematicsChapter 7: Vectors in Plane
Unit Vectors
A unit vector has length 1.
Learning objectives
- 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.
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 other way is -u/|u|.
Mathematical form
u = 6i + 8j.
j = (35)i + (45)j check
Where / symbols
- 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 (3/5)i+(4/5)j. Check: (3/5)^2+(4/5)^2=1. The unit vector the other way is -u/|u|.
- SAME DIRECTION
- u = 6i+8j, |u|=10, unit = (6/10)i+(8/10)j = (3/5)i+(4/5)j. Do not skip simplifying 6/10.
- 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.
Overview
Divide by the length:
Same Direction
u = 6i+8j, |u|=10, unit = (610)i+(810)j = (35)i+(45)j. Do not skip simplifying 610.
Examples |6i+8j|=10 · 3/5 and 4/5 · 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.
Examples i and j are unit · -i is unit the other way · Not 6i+8j itself
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.
Worked example
- 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
In simple words
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|.
Key points
- u / |u|
- length 1
- divide each component
- 0 has no direction unit
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Unit Vectors
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- 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 Misconception
- 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|.
FBISE Exam Check
- 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|.
60-second revision
- 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
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