More with Vectors

On the first vectors page a bunny hopped: every hop had a magnitude and a direction, and chaining hops tip-to-tail added the vectors. Here we keep working with that same hop, but stretch it, flip it, break it into unit-sized steps, and finally attach the physical units that turn an abstract arrow into a displacement, a velocity, or a force.

Scaling and Negation

Multiplying a vector by an ordinary number — a scalar — stretches or shrinks it without changing the line it points along. Each component is multiplied by the scalar:

The magnitude scales by the absolute value of the scalar,

so $2\vec{a}$ is twice as long and $\tfrac{1}{2}\vec{a}$ is half as long, both pointing the same way as $\vec{a}$.

A negative scalar does something extra: it reverses the direction. The special case $c = -1$ is negation, $-\vec{a} = (-1)\vec{a}$, which is the same-length hop pointing the opposite way. This is exactly what lets us subtract: $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$.

Unit Vectors

A unit vector is any vector with magnitude $1$. Two of them are special enough to get their own names. Let the bunny’s single hop one space to the right be

and its single hop one space up be

(The “hat” is read “i-hat” and “j-hat,” and always marks a unit vector.) Using the scaling rule from the previous section, any vector is just a scaled right-hop plus a scaled up-hop:

So $\langle 3, 2 \rangle$ is “three hops right and two hops up,” or $3\hat{\imath} + 2\hat{\jmath}$. Every press of an arrow key below adds one unit vector to the bunny’s running displacement from where its current carrot appeared. Each hop is drawn as a blue vector; reach the carrot and their green sum is the total displacement — the same head-to-tail addition from the first page, now built one unit vector at a time.

Vectors Carry Units

So far the bunny’s hops have been pure numbers on a grid. In physics a vector almost always carries a unit that tells you what it measures. The bunny’s hop is a displacement, so its components have units of length — say meters — and a hop of $\langle 3, 2 \rangle\ \text{m}$ means three meters east and two meters north. Displacement is the most intuitive vector precisely because you can pace it out.

The power of the vector formalism is that the algebra never changes. Adding components, scaling, and decomposing into $\hat{\imath}$ and $\hat{\jmath}$ work identically no matter what the arrow represents. The units simply ride along inside the magnitude:

Scaling even respects the units. Multiplying a velocity $\vec{v}$ (in $\text{m/s}$) by a time $t$ (a scalar in $\text{s}$) gives a displacement in meters, $\vec{d} = \vec{v}\,t$ — the seconds cancel just as they would in ordinary arithmetic, while the direction of the velocity is preserved. Every tool you built on the first page now applies to forces on a free-body diagram, to electric fields around a charge, and to any other quantity with both a magnitude and a direction.