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Calculus IIIGrades 9–12University3 min read

Dot Product

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

Cheat sheet
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.

Orthogonality

Two nonzero vectors are perpendicular exactly when

$$ \vec u\cdot\vec v=0. $$

This provides an algebraic perpendicularity test without needing a diagram. The zero vector has zero dot product with everything but has no direction, so it is handled separately in geometric statements.

Scalar projection

The signed scalar projection of $\vec u$ onto $\vec v$ is

$$ \operatorname{comp}_{\vec v}\vec u =\frac{\vec u\cdot\vec v}{\|\vec v\|}. $$

It measures how much of $\vec u$ points along the direction of $\vec v$. A negative value points opposite that direction.

Vector projection

The vector projection is

$$ \operatorname{proj}_{\vec v}\vec u =\frac{\vec u\cdot\vec v}{\|\vec v\|^2}\vec v. $$

Subtracting this projection from $\vec u$ gives a component perpendicular to $\vec v$. Together, the parallel and perpendicular components reconstruct $\vec u$.

Work

For a constant force $\vec F$ causing displacement $\vec d$, work is

$$ W=\vec F\cdot\vec d =\|\vec F\|\|\vec d\|\cos\theta. $$

Only the force component along the displacement contributes. Perpendicular force does no work under this idealized definition. Units are joules when force is in newtons and displacement in metres.

Useful properties

The dot product is commutative and distributive:

$$ \vec u\cdot\vec v=\vec v\cdot\vec u, $$
$$ \vec u\cdot(\vec v+\vec w)=\vec u\cdot\vec v+\vec u\cdot\vec w. $$

Also $\vec v\cdot\vec v=\|\vec v\|^2$.

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?
Check your understanding

Try it yourself

Hints are part of learning. Open one whenever it makes the next step feel possible.

1 practice question
Question 1Calculate a dot product · Gentle

Find ⟨2, −1, 3⟩ · ⟨1, 4, −2⟩.

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