The dot product, magnitude and angle in ℝ²

1How much of a push really counts

Push a heavily loaded wooden scooter up a village road and you lean on its handle, so your push points partly along the road and partly down into the ground. Only the part along the road moves the load forward. Physics measures the useful effort by the work done: the component of the force along the motion, times the distance moved. For a force of size $FF$ at angle $\theta \backslash theta$ to a displacement of length $dd$, the work is $W=Fd\cos \theta W\; =\; F\; d\; \backslash cos\backslash theta$.

So far our vector space ${\mathbb{R}}^2\backslash mathbb\{R\}^2$ has had no lengths and no angles at all: the ten rules never mention them. This lesson adds one more operation, the dot product, which produces lengths, angles, perpendicularity, and the work formula in a single line of arithmetic.

2Properties of the dot product

For all $\vec{u},\vec{v},\vec{w}\in {\mathbb{R}}^2\backslash vec\{u\},\; \backslash vec\{v\},\; \backslash vec\{w\}\; \backslash in\; \backslash mathbb\{R\}^2$ and every real $kk$:

The vector space ${\mathbb{R}}^2\backslash mathbb\{R\}^2$ together with the dot product is called the Euclidean vector space ${\mathbb{R}}^2\backslash mathbb\{R\}^2$: the vector space in which the distances and angles of Euclid's geometry can be measured.

3The angle between two vectors

Cauchy–Schwarz says that for non-zero vectors $-1\le \frac{\vec{u}\cdot \vec{v}}{\mathrm{\parallel }\vec{u}\mathrm{\parallel }\mathrm{\parallel }\vec{v}\mathrm{\parallel }}\le 1-1\; \backslash le\; \backslash frac\{\backslash vec\{u\}\; \backslash cdot\; \backslash vec\{v\}\}\{\backslash |\backslash vec\{u\}\backslash |\backslash ,\backslash |\backslash vec\{v\}\backslash |\}\; \backslash le\; 1$, so this number is the cosine of exactly one angle between $0^{\circ }0^\backslash circ$ and ${180}^{\circ }180^\backslash circ$. It is the angle you see in the diagram. Apply the cosine rule to the triangle with sides $\vec{u}\backslash vec\{u\}$, $\vec{v}\backslash vec\{v\}$ and $\vec{u}-\vec{v}\backslash vec\{u\}\; -\; \backslash vec\{v\}$:

$$\mathrm{\parallel }\vec{u}-\vec{v}{\mathrm{\parallel }}^2=\mathrm{\parallel }\vec{u}{\mathrm{\parallel }}^2+\mathrm{\parallel }\vec{v}{\mathrm{\parallel }}^2-2\mathrm{\parallel }\vec{u}\mathrm{\parallel }\mathrm{\parallel }\vec{v}\mathrm{\parallel }\cos \theta \mathrm{.}\backslash |\backslash vec\{u\}\; -\; \backslash vec\{v\}\backslash |^2\; =\; \backslash |\backslash vec\{u\}\backslash |^2\; +\; \backslash |\backslash vec\{v\}\backslash |^2\; -\; 2\backslash |\backslash vec\{u\}\backslash |\backslash ,\backslash |\backslash vec\{v\}\backslash |\; \backslash cos\backslash theta\; .$$

Expanding the left side with the dot product gives $\mathrm{\parallel }\vec{u}{\mathrm{\parallel }}^2-2\vec{u}\cdot \vec{v}+\mathrm{\parallel }\vec{v}{\mathrm{\parallel }}^2\backslash |\backslash vec\{u\}\backslash |^2\; -\; 2\backslash vec\{u\}\; \backslash cdot\; \backslash vec\{v\}\; +\; \backslash |\backslash vec\{v\}\backslash |^2$. Compare the two:

$$\vec{u}\cdot \vec{v}=\mathrm{\parallel }\vec{u}\mathrm{\parallel }\mathrm{\parallel }\vec{v}\mathrm{\parallel }\cos \theta ,\cos \theta =\frac{\vec{u}\cdot \vec{v}}{\mathrm{\parallel }\vec{u}\mathrm{\parallel }\mathrm{\parallel }\vec{v}\mathrm{\parallel }},0^{\circ }\le \theta \le {180}^{\circ }\mathrm{.}\backslash vec\{u\}\; \backslash cdot\; \backslash vec\{v\}\; =\; \backslash |\backslash vec\{u\}\backslash |\backslash ,\backslash |\backslash vec\{v\}\backslash |\; \backslash cos\backslash theta,\; \backslash qquad\; \backslash cos\backslash theta\; =\; \backslash frac\{\backslash vec\{u\}\; \backslash cdot\; \backslash vec\{v\}\}\{\backslash |\backslash vec\{u\}\backslash |\backslash ,\backslash |\backslash vec\{v\}\backslash |\},\; \backslash quad\; 0^\backslash circ\; \backslash le\; \backslash theta\; \backslash le\; 180^\backslash circ.$$

For the diagram, $\cos \theta =\frac{14}{\sqrt{26}\sqrt{20}}\approx 0.614\backslash cos\backslash theta\; =\; \backslash frac\{14\}\{\backslash sqrt\{26\}\backslash ,\backslash sqrt\{20\}\}\; \backslash approx\; 0.614$, so $\theta \approx {52.1}^{\circ }\backslash theta\; \backslash approx\; 52.1^\backslash circ$. Because the lengths are positive, the sign of the dot product tells you the type of angle at once:

4Work done by a force

In physics the work done by a constant force $\vec{F}\backslash vec\{F\}$ (in newtons) while its point of application moves through a displacement $\vec{d}\backslash vec\{d\}$ (in metres) is

$$W=\vec{F}\cdot \vec{d}=\mathrm{\parallel }\vec{F}\mathrm{\parallel }\mathrm{\parallel }\vec{d}\mathrm{\parallel }\cos \theta \text{joules}\mathrm{.}W\; =\; \backslash vec\{F\}\; \backslash cdot\; \backslash vec\{d\}\; =\; \backslash |\backslash vec\{F\}\backslash |\backslash ,\backslash |\backslash vec\{d\}\backslash |\; \backslash cos\backslash theta\; \backslash quad\; \backslash text\{joules\}.$$

The factor $\mathrm{\parallel }\vec{F}\mathrm{\parallel }\cos \theta \backslash |\backslash vec\{F\}\backslash |\; \backslash cos\backslash theta$ is the scalar projection of $\vec{F}\backslash vec\{F\}$ on the direction of motion: the green part of the force in the diagram. The vertical part of the pull does no work, because it is perpendicular to the motion.