Dot Product

The dot product combines two vectors into a scalar that measures directional alignment, angle, projection, and work.

Three coordinates describe independent movement along three perpendicular axes. A point in space is connected to the coordinate planes by dashed projections. Component arrows build its position one axis at a time.

The dot product is positive for broadly aligned vectors, zero for perpendicular vectors, and negative for opposing vectors.

Component definition

For vectors in three dimensions,

\vec u=\langle u_1,u_2,u_3\rangle,qquad \vec v=\langle v_1,v_2,v_3\rangle,

their dot product is

\vec u\cdot\vec v=u_1v_1+u_2v_2+u_3v_3.

In two dimensions, omit the third component. The output is a scalar, not a vector.

Geometric definition

If $\theta$ is the angle between nonzero vectors,

\vec u\cdot\vec v=\|\vec u\|\|\vec v\|\cos\theta.

This connects component algebra to direction. The angle between vectors is conventionally chosen from $0$ to $\pi$.

Worked example: compute a dot product

The sign gives useful qualitative information before the angle is calculated.

Finding an angle

For nonzero vectors,

\cos\theta=\frac{\vec u\cdot\vec v}{\|\vec u\|\|\vec v\|}.

Compute the dot product and both magnitudes, then use inverse cosine. Keep full precision before evaluating the angle and confirm the sign agrees with acute, right, or obtuse geometry.

Common mistakes

Returning a component vector. The dot product is one scalar.

Multiplying magnitudes without cosine. Direction matters.

Using inverse tangent for the angle between arbitrary vectors. Use inverse cosine of the normalized dot product.

Dividing vector projection by $\|v\|$ instead of $\|v\|^2$. Distinguish scalar and vector projection.

Rounding before inverse cosine. The angle can shift noticeably.

Quick self-check

Were corresponding components multiplied and then added?
Is the output a scalar?
Does the sign match the expected angle type?
Are both vectors nonzero before finding an angle?
Is scalar versus vector projection distinguished?
Do application units and direction interpretations make sense?