Vectors (2D)

Components, magnitude, and the dot product in the plane.

The idea

A vector is a quantity carrying both a size and a direction — a force, a velocity, a displacement — and we picture it as an arrow. Only the arrow's length and direction matter, not where it is drawn, so the two displacements from its tail to its head describe it completely. These are its components, written $\mathbf{v} = \langle a, b \rangle$: $a$ units across and $b$ units up. The vector from a point $P$ to a point $Q$ therefore has components equal to $Q s coordinates minus $P s, and adding two vectors adds their components — the same result as placing the second arrow's tail at the first arrow's head.

The components are the legs of a right triangle whose hypotenuse is the arrow, which gives the vector's length.

Definition (Magnitude and dot product).

The magnitude of $\mathbf{v} = \langle a, b \rangle$ is its length, $|\mathbf{v}| = \sqrt{a^{2} + b^{2}}.$ The dot product of $\mathbf{u} = \langle u_{1}, u_{2} \rangle$ and $\mathbf{v} = \langle v_{1}, v_{2} \rangle$ is the number $\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2}.$

The dot product multiplies matching components and adds them, returning a number rather than a vector. The formula is pure arithmetic, yet it knows the angle between the arrows.

Proposition.

If $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$, then $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}|\,|\mathbf{v}|\cos\theta.$

So the dot product computes angles from components alone.

Ways to work on it

• Walkthrough. Read off a vector's components, compute its magnitude, and take a dot product.
• Practice. Find the magnitude of a vector from its components.
• Hardest. Use orthogonality to recover a component, then find an angle.

Not sure where to start? Take the ten-question placement test.