3.3 · Vectors, dot products, and parametric curves

Components describe a directed displacement

For a vector $\displaystyle \langle a,b\rangle$, magnitude is $\displaystyle \sqrt{a^2+b^2}$. Addition and scalar multiplication act on components. A unit vector in its direction is the original vector divided by its nonzero magnitude.

Two equal vectors can have different starting locations. A component diagram distinguishes a vector’s direction and length from the position of an endpoint.

The dot product compares directions

For $\displaystyle u=\langle a,b\rangle$ and $\displaystyle v=\langle c,d\rangle$, the dot product is $\displaystyle ac+bd$. For nonzero vectors it also equals $\displaystyle |u||v|\cos\theta$. A zero dot product therefore identifies perpendicular vectors.

The zero vector is orthogonal in the algebraic sense but has no defined direction angle. Check nonzero magnitudes before dividing to solve for an angle.

$$u\cdot v=|u||v|\cos\theta$$

Parametric equations retain a time or path variable

For $\displaystyle x=t+1$, $\displaystyle y=t^2$, eliminate $\displaystyle t$ using $\displaystyle t=x-1$ to obtain $\displaystyle y=(x-1)^2$. The original parameter interval may describe only part of this parabola.

Elimination alone does not preserve orientation. Record how points move as $\displaystyle t$ increases, and translate any restrictions on $\displaystyle t$ into restrictions on the rectangular curve.