Dot Product in Component Form

Dot Product in Component Form

What is it?

i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.

In simple words

• 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.

Important terms

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.

What you should learn

• 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.

Examples

• <1,2,2>·<2,1,-1>=2
• |a|² = a·a
• Not a vector

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.

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.

Full school-level explanation

i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.

Understand the idea, then apply it:

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.

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.

Remember

• sum ai bi

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.

Remember

• focused answer
• accurate support

Examples

• State the central idea
• Use one accurate detail
• Check the final response

How to answer a dot product in component form question

How it works

• 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.

Remember

• sum ai bi
• a·b = |a||b| cos theta
• also sum of products of components
• scalar result
• sum ai bi
• focused answer
• accurate support
• sum ai bi

Deeper understanding

Go beyond memorising Dot Product in Component Form. Connect the definition to reasons, relationships, exceptions, and exam wording.

Common mistakes

Exam focus

Core Idea

What to Know

Example or Evidence

How to Answer

Exam focus

Quick revision

• sum ai bi
• a·b = |a||b| cos theta
• also sum of products of components
• scalar result
• 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.

Dot product is a number. Do not write a·b as a vector.

For long answers on Dot Product in Component Form: define clearly → explain the main parts → give one example → state one exam caution.