The Dot Product

Multiply two arrows, get a number: how much one points along the other.

12 minUpdated 21 October 2026scalar productprojectionworkorthogonality

Lesson at a glanceShow the key points

a·b = ab cos θ is a scalar: the length of one vector times the projection of the other on it.
a·b = 0 for non-zero vectors means they are perpendicular; it is negative for obtuse angles.
The dot product is commutative and distributive.
In components, a·b = aₓbₓ + a_yb_y + a_zb_z, since î·î = 1 and î·ĵ = 0.
Work is a dot product, W = F·r, so it can be positive, negative or zero.

Two arrows go in, a single number comes out. The dot product measures how much two vectors agree about direction.

Definition

The dot product (or scalar product) of $\vec a$ and $\vec b$ is

Dot product(2.4)

\vec a\cdot\vec b = ab\cos\theta

where θ is the angle between the vectors when drawn tail to tail.

Read it as “the length of $\vec a$ times the shadow of $\vec b$ on $\vec a$ ” (or the other way round, since the formula is symmetric). Explore it:

Interactive

The dot product as a shadow

Drag either arrowhead. The thick gold segment is the shadow of b on the line of a. Its signed length, times |a|, is a · b.

Acute angle: b leans along a, so a · b is positive.

a

4.12

b

3.35

\theta

49.4°

b\cos\theta

2.18

\vec a\cdot\vec b

9.00

Three cases are worth fixing in memory:

Same direction (θ = 0): $\vec a\cdot\vec b = ab$ , the largest possible value.
Perpendicular (θ = 90°): $\cos 90^\circ = 0$ , so the dot product is zero. This is the standard test for perpendicularity.
Opposite (θ = 180°): $\vec a\cdot\vec b = -ab$ .

Algebraic rules

The dot product behaves like ordinary multiplication in two important ways:

\vec a\cdot\vec b = \vec b\cdot\vec a \qquad\text{(commutative)}

\vec a\cdot(\vec b + \vec c) = \vec a\cdot\vec b + \vec a\cdot\vec c \qquad\text{(distributive)}

Worked example

Negative work

A crane pulls a crate upward with a force of 15 N, but the crate still moves 3.0 m downward. How much work does the crane’s force do?

The dot product from components

Write $\vec a = a_x\hat\imath + a_y\hat\jmath + a_z\hat k$ and similarly for $\vec b$ . Expanding the product with the distributive law gives nine terms, each a dot product of two unit vectors. Those are easy, because the unit vectors have length 1 and are mutually perpendicular:

·	$\hat\imath$	$\hat\jmath$	$\hat k$
$\hat\imath$	1	0	0
$\hat\jmath$	0	1	0
$\hat k$	0	0	1

Six of the nine terms vanish, leaving a formula you will use constantly:

Dot product in components

\vec a\cdot\vec b = a_xb_x + a_yb_y + a_zb_z

Worked example

Angle between two vectors

Find the angle between $\vec a = 2\hat\imath + 2\hat\jmath - \hat k$ and $\vec b = 6\hat\imath - 3\hat\jmath + 2\hat k$ .

What to remember

1 $\vec a\cdot\vec b = ab\cos\theta$ is a scalar: length of one × shadow of the other.
2Zero dot product (for non-zero vectors) means perpendicular.
3In components: $a_xb_x + a_yb_y + a_zb_z$ . Use it to find angles.
4Work is a dot product, so it can be positive, negative or zero.

Check yourself

For which value of p are 3î + pĵ and 2î − 6ĵ perpendicular?

A person carries a bag at constant height while walking horizontally. The work done by the upward holding force is…

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PreviousComponents & Unit Vectors Next The Cross Product

The Dot Product

Multiply two arrows, get a number: how much one points along the other.

12 minUpdated 21 October 2026scalar productprojectionworkorthogonality

Lesson at a glanceShow the key points

a·b = ab cos θ is a scalar: the length of one vector times the projection of the other on it.
a·b = 0 for non-zero vectors means they are perpendicular; it is negative for obtuse angles.
The dot product is commutative and distributive.
In components, a·b = aₓbₓ + a_yb_y + a_zb_z, since î·î = 1 and î·ĵ = 0.
Work is a dot product, W = F·r, so it can be positive, negative or zero.

Two arrows go in, a single number comes out. The dot product measures how much two vectors agree about direction.

Definition

The dot product (or scalar product) of $\vec a$ and $\vec b$ is

Dot product(2.4)

\vec a\cdot\vec b = ab\cos\theta

where θ is the angle between the vectors when drawn tail to tail.

Read it as “the length of $\vec a$ times the shadow of $\vec b$ on $\vec a$ ” (or the other way round, since the formula is symmetric). Explore it:

Interactive

The dot product as a shadow

Drag either arrowhead. The thick gold segment is the shadow of b on the line of a. Its signed length, times |a|, is a · b.

Acute angle: b leans along a, so a · b is positive.

a

4.12

b

3.35

\theta

49.4°

b\cos\theta

2.18

\vec a\cdot\vec b

9.00

Three cases are worth fixing in memory:

Same direction (θ = 0): $\vec a\cdot\vec b = ab$ , the largest possible value.
Perpendicular (θ = 90°): $\cos 90^\circ = 0$ , so the dot product is zero. This is the standard test for perpendicularity.
Opposite (θ = 180°): $\vec a\cdot\vec b = -ab$ .

Algebraic rules

The dot product behaves like ordinary multiplication in two important ways:

\vec a\cdot\vec b = \vec b\cdot\vec a \qquad\text{(commutative)}

\vec a\cdot(\vec b + \vec c) = \vec a\cdot\vec b + \vec a\cdot\vec c \qquad\text{(distributive)}

Worked example

Negative work

A crane pulls a crate upward with a force of 15 N, but the crate still moves 3.0 m downward. How much work does the crane’s force do?

The dot product from components

·	$\hat\imath$	$\hat\jmath$	$\hat k$
$\hat\imath$	1	0	0
$\hat\jmath$	0	1	0
$\hat k$	0	0	1

Six of the nine terms vanish, leaving a formula you will use constantly:

Dot product in components

\vec a\cdot\vec b = a_xb_x + a_yb_y + a_zb_z

Worked example

Angle between two vectors

Find the angle between $\vec a = 2\hat\imath + 2\hat\jmath - \hat k$ and $\vec b = 6\hat\imath - 3\hat\jmath + 2\hat k$ .

What to remember

1 $\vec a\cdot\vec b = ab\cos\theta$ is a scalar: length of one × shadow of the other.
2Zero dot product (for non-zero vectors) means perpendicular.
3In components: $a_xb_x + a_yb_y + a_zb_z$ . Use it to find angles.
4Work is a dot product, so it can be positive, negative or zero.

Check yourself

For which value of p are 3î + pĵ and 2î − 6ĵ perpendicular?

A person carries a bag at constant height while walking horizontally. The work done by the upward holding force is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousComponents & Unit Vectors Next The Cross Product

Definition#

Algebraic rules#

The dot product from components#

Check yourself

Keep exploring

Definition#

Algebraic rules#

The dot product from components#

Check yourself

Keep exploring

Definition

Algebraic rules

The dot product from components

Definition

Algebraic rules

The dot product from components