MathematicsChapter 3: Vectors
Dot Product in Component Form
i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...).
Learning objectives
- State the rule or meaning of dot product in component form in easy words.
- Apply dot product in component form in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Definition
i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.
Formula
j=0. Expand (x1 i+...)
Where / symbols
- DOT PRODUCT IN COMPONENT FORM
- i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.
- COMPONENTS
- i·i=1, i·j=0.
Relationship
- a·b = |a||b| cos theta
Overview
Understand the idea, then apply it:
Main Idea
i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.
Examples <1,2,2>·<2,1,-1>=2 · |a|^2 = a·a · Not a vector
Exam Use
Write the rule first: i·i=1, i·j=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Examples State the central idea · Use one accurate detail · Check the final response
WHAT — DOT PRODUCT IN COMPONENT FORM
i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.
WHY — Why it matters
This rule is a separate learning unit inside dot (scalar) product. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: i·i=1, i·j=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
How to answer a dot product in component form question
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
In simple words
Dot product is a number. Do not write a·b as a vector.
Key points
- sum ai bi
- a·b = |a||b| cos theta
- also sum of products of components
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Dot Product in Component Form
Sticky Notes · Quick Revision
Core Idea
- i·i=1, i·j=0.
What to Know
- i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.
Example or Evidence
- <1,2,2>·<2,1,-1>=2
- |a|2 = a·a
- Not a vector
How to Answer
- Write the rule first: i·i=1, i·j=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Common Error
- Dot product is a number. Do not write a·b as a vector.
FBISE Exam Check
- State the rule, show the working, and check the answer.
60-second revision
- i·i=1, i·j=0.
- <1,2,2>·<2,1,-1>=2
- |a|2 = a·a
- State the rule, show the working, and check the answer.
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