The dot product

A vector operation that gives a scalar

The dot product is a way to multiply two vectors and get a scalar result. It is written

ec{A}cdotec{B}

The dot product measures how much two vectors point in the same direction. If they point mostly the same way, the dot product is positive. If they are perpendicular, it is zero. If they point mostly opposite ways, it is negative.

The geometric definition is

ec{A}cdotec{B} = |ec{A}||ec{B}|cos heta

where $heta$ is the angle between the vectors.

Component form

In two dimensions, the dot product can be calculated from components:

ec{A}cdotec{B} = A_xB_x + A_yB_y

In three dimensions:

ec{A}cdotec{B} = A_xB_x + A_yB_y + A_zB_z

This form is often easier when components are known. The geometric form is often easier when magnitudes and the angle are known.

Projection

The dot product is closely related to projection. A projection tells how much of one vector lies along another direction. If you want the component of ec{A} along the direction of ec{B}, the dot product helps calculate it.

The factor |ec{A}|cos heta is the part of ec{A} parallel to ec{B}. This is why cosine appears in the dot product. It selects the aligned part.

Work as a dot product

One of the most important physics uses of the dot product is work. When a constant force moves an object through a displacement,

W = ec{F}cdotec{d}

Using the geometric definition,

$W = Fdcos heta$

Only the component of force in the direction of displacement does work. If you pull a suitcase forward with an upward-angled handle, only the forward component contributes to work on the suitcase's forward motion.

If force and displacement are in the same direction, $heta = 0$ and $cos 0 = 1$, so $W = Fd$. If they are perpendicular, $heta = 90^circ$ and $cos 90^circ = 0$, so no work is done by that force. If they are opposite, work is negative.

Finding angles between vectors

The dot product can also find the angle between vectors:

cos heta = rac{ec{A}cdotec{B}}{|ec{A}||ec{B}|}

Then

ight)$$ This is useful in mechanics, geometry, computer graphics, and fields. ## Orthogonality Two vectors are orthogonal if their dot product is zero. Orthogonal means perpendicular in the vector sense. This idea is central in physics because independent directions are often represented by orthogonal axes. The unit vectors $hat{i}$, $hat{j}$, and $hat{k}$ are mutually orthogonal. ## Dot product versus ordinary multiplication The dot product is not the same as multiplying magnitudes. Direction matters. Two large vectors can have a dot product of zero if they are perpendicular. A force can be large but do no work if it acts perpendicular to displacement. ## The big idea The dot product measures alignment. It turns two vectors into a scalar by selecting how much they point in the same direction. In physics, it explains work, projections, angles, and perpendicularity. Whenever a physical effect depends on the component of one vector along another, the dot product is likely involved.