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Dot (scalar) Product
a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school). Also a · b = a1 b1 + a2 b2 + a3 b3. Result is a scalar. a · a = |a|^2. Perpendicular: a · b = 0 (if neither is 0). Parallel: angle 0 or 180, |a · b| = |a||b|. Meaning: the dot asks 'how much of a points along b?' Work = F · d in physics.
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Dot (scalar) Product
What is it?
a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school). Also a · b = a1 b1 + a2 b2 + a3 b3. Result is a scalar. a · a = |a|^2. Perpendicular: a · b = 0 (if neither is 0). Parallel: angle 0 or 180, |a · b| = |a||b|. Meaning: the dot asks 'how much of a points along b?' Work = F · d in physics.
In simple words
- a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school).
- Also a · b = a1 b1 + a2 b2 + a3 b3.
- Result is a scalar.
- Dot product is a number. Do not write a·b as a vector.
Important terms
DOT (SCALAR) PRODUCT: a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school). Also a · b = a1 b1 + a2 b2 + a3 b3. Result is a scalar. a · a = |a|^2. Perpendicular: a · b = 0 (if neither is 0). Parallel: angle 0 or 180, |a · b| = |a||b|. Meaning: the dot asks 'how much of a points along b?' Work = F · d in physics.
1. GEOMETRY: cos theta = a·b|a||b|. Acute if dot > 0, obtuse if dot < 0, right if 0. Projection of a on b is |a| cos theta along b.
2. COMPONENTS: i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.
What you should learn
- State the main ideas and notation used in dot (scalar) product.
- Apply the rules of dot (scalar) product in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- Work F·d
- perp: dot 0
- theta from cos-1
GEOMETRY
cos theta = a·b|a||b|. Acute if dot > 0, obtuse if dot < 0, right if 0. Projection of a on b is |a| cos theta along b.
COMPONENTS
i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.
Recommended for Class 11
Full school-level explanation
a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school). Also a · b = a1 b1 + a2 b2 + a3 b3. Result is a scalar. a · a = |a|^2. Perpendicular: a · b = 0 (if neither is 0). Parallel: angle 0 or 180, |a · b| = |a||b|. Meaning: the dot asks 'how much of a points along b?' Work = F · d in physics.
Angle, then components:
WHAT — DOT (SCALAR) PRODUCT
a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school). Also a · b = a1 b1 + a2 b2 + a3 b3. Result is a scalar. a · a = |a|^2. Perpendicular: a · b = 0 (if neither is 0). Parallel: angle 0 or 180, |a · b| = |a||b|. Meaning: the dot asks 'how much of a points along b?' Work = F · d in physics.
RULE — 1. GEOMETRY
cos theta = a·b|a||b|. Acute if dot > 0, obtuse if dot < 0, right if 0. Projection of a on b is |a| cos theta along b.
METHOD — 2. COMPONENTS
i·i=1, i·j=0. Expand (x1 i+...)·(y1 i+...). Distributive: a·(b+c)=a·b+a·c. Commutative: a·b=b·a.
WORKED EXAMPLE
Worked: angle between a=<1,0,0> and b=<1,1,0>. a·b = 1 |a|=1, |b|=sqrt(2) cos theta = 1/sqrt(2) theta = 45 degrees (or pi/4)
GEOMETRY
cos theta = a·b|a||b|. Acute if dot > 0, obtuse if dot < 0, right if 0. Projection of a on b is |a| cos theta along b.
Remember
- cos from the formula
Examples
- Work F·d
- perp: dot 0
- theta from cos-1
COMPONENTS
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
How to answer a dot (scalar) product question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- a·b = |a||b| cos theta
- also sum of products of components
- 45 degrees
- scalar result
- cos from the formula
- sum ai bi
- a·b = |a||b| cos theta
- also sum of products of components
How the ideas connect
Worked: angle between a=<1,0,0> and b=<1,1,0>. a·b = 1 |a|=1, |b|=sqrt(2) cos theta = 1/sqrt(2) theta = 45 degrees (or pi/4)
Deeper understanding
Go beyond memorising Dot (scalar) Product. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: angle between a=<1,0,0> and b=<1,1,0>. a·b = 1 |a|=1, |b|=sqrt(2) cos theta = 1/sqrt(2) theta = 45 degrees (or pi/4)
Common mistakes
Exam focus
Core Idea
1. GEOMETRY
2. COMPONENTS
Worked Example
Exam focus
Quick revision
- a·b = |a||b| cos theta
- also sum of products of components
- 45 degrees
- scalar result
- DOT (SCALAR) PRODUCT: a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school). Also a · b = a1
- 1. GEOMETRY: cos theta = a·b|a||b|. Acute if dot > 0, obtuse if dot < 0, right if 0. Projection of a on b
Dot product is a number. Do not write a·b as a vector.
For long answers on Dot (scalar) Product: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- a·b = |a||b| cos theta
- also sum of products of components
- 45 degrees
- scalar result
Important formulas
- a·b = |a||b| cos theta
- also sum of products of components
- 45 degrees
Common exam traps
- Dot product is a number. Do not write a·b as a vector.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Dot product is a number. Do not write a·b as a vector.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
Dot product is a number. Do not write a·b as a vector.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
a·b = |a||b| cos theta
also sum of products of components
45 degrees
scalar result
a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school). Also a · b = a1 b1 + a2 b2 + a3 b3. Result is a scalar. a · a = |a|^2. Perpe
a·b = |a||b| cos theta
- Definition
- a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school). Also a · b = a1 b1 + a2 b2 + a3 b3. Result is a scalar. a · a = |a|^2. Perpe
- Formula
- a·b = |a||b| cos theta
- Example
- Work F·d
- Critical warning
- Dot product is a number. Do not write a·b as a vector.
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Exam Practice
Define
Q1. Define Dot (scalar) Product.
Explain
Q2. Explain: GEOMETRY.
Example
Q3. Give one clear example of Dot (scalar) Product and say why it fits.
Mistake
Q4. State one common mistake students make about Dot (scalar) Product.
Exam note
Q5. Write a short exam note on Dot (scalar) Product: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.