The Dot Product

The boards question that opened class looked innocent: what should multiplying two vectors even mean? Numbers multiply into numbers — but arrows have direction, and every group’s proposal had to decide what direction does to a product. The dot product is the answer mathematics settled on, and its defining choice is bold: two vectors in, one scalar out. The directions do not survive; they get measured.

$\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$

where $\theta$ is the angle between the vectors. The $\cos \theta$ is the measuring device: it rewards vectors for pointing the same way, ignores the part of each vector that is perpendicular to the other. Pulling a wagon, only the part of your pull along the ground moves the wagon forward — the dot product of force and displacement is exactly the work that gets done, and raising the handle shrinks it.

Two formulas, one number

The geometric formula needs the angle. The miracle is that components compute the same number without ever finding it:

$\vec{a}\cdot \vec{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$

Multiply matching coordinates, add. For $(1,2,2)$ and $(2,2,1)$: $2+4+2=8$. Set the two formulas equal and the angle falls out — $\cos \theta =\frac{8}{3\times 3}$, so $\theta \approx 27°$. An angle in three-space, measured without a protractor, in a space nobody can fully draw. This is the trick the rest of the unit runs on.

What the sign tells you

Because $|\vec{a}|$ and $|\vec{b}|$ are positive, the sign of the dot product belongs entirely to $\cos \theta$:

$\vec{a}\cdot \vec{b}$	Angle between	Meaning
positive	less than $90°$	broadly agreeing directions
zero	exactly $90°$	perpendicular — orthogonal
negative	more than $90°$	broadly opposed directions

The middle row does the most work in this course: a dot product of zero is a perpendicularity test, no picture required. It is how Equations of Planes will define a plane with one vector.

The properties got the True or False treatment before anyone trusted them: commutative — true, both formulas are symmetric; $\vec{a}\cdot \vec{a}$ — always $|\vec{a}{|}^2$, a vector dotted with itself squares its own length; associative — not even a well-formed question, since $\vec{a}\cdot \vec{b}$ is a scalar and cannot dot anything further. Knowing why that last one fails is worth more than ten computations. Dot and Cross Product Practice covers angles, work, and projections; the product that keeps direction instead of measuring it is The Cross Product.

Curriculum connection

C2.4

perform the operation of dot product on two vectors represented as directed line segments (i.e., using $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$) and in Cartesian form (i.e., using $\vec{a}\cdot \vec{b}=a_1{b}_1+a_2{b}_2$ or $\vec{a}\cdot \vec{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$) in two-space and three-space, and describe applications of the dot product (e.g., determining the angle between two vectors; determining the projection of one vector onto another) Sample problem: Describe how the dot product can be used to compare the work done in pulling a wagon over a given distance in a specific direction using a given force for different positions of the handle.

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C2.5

determine, through investigation, properties of the dot product (e.g., investigate whether it is commutative, distributive, or associative; investigate the dot product of a vector with itself and the dot product of orthogonal vectors) Sample problem: Investigate geometrically and algebraically the relationship between the dot product of the vectors $(1,0,1)$ and $(0,1,-1)$ and the dot product of scalar multiples of these vectors. Does this relationship apply to any two vectors? Find a vector that is orthogonal to both the given vectors.

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C2.8

solve problems involving dot product and cross product (e.g., determining projections, the area of a parallelogram, the volume of a parallelepiped), including problems arising from real-world applications (e.g., determining work, torque, ground speed, velocity, force) Sample problem: Investigate the dot products $\vec{a}\cdot (\vec{a}\times \vec{b})$ and $\vec{b}\cdot (\vec{a}\times \vec{b})$ for any two vectors $\vec{a}$ and $\vec{b}$ in three-space. What property of the cross product $\vec{a}\times \vec{b}$ does this verify?

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