MathematicsChapter 3: Vectors

Dot (scalar) Product

a · b = |a| |b| cos theta, where theta is the angle between them (0 to 180 school).

Learning objectives

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.

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. 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.

Mathematical form

theta = 45 degrees (or pi/4)

b = a1 b1 + a2 b2 + a3 b3. Result is a scala

b = 0 (if neither is 0). Parallel

Work = F

Where / symbols

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.

Relationship

a·b = |a||b| cos theta

Overview

Angle, then components:

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.

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.

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

WHAT — DOT (SCALAR) PRODUCT

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)

How to answer a dot (scalar) product question

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.

Worked example

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)

In simple words

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

Key points

a·b = |a||b| cos theta
also sum of products of components
cos from the formula
sum ai bi

Exam preparation

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