Components & Unit Vectors

Shine a light straight down on an arrow and look at its shadow. That shadow is a component, and components turn vector geometry into plain arithmetic.

Resolving a vector in a plane#

Put a vector $\vec a = \vec{OA}$ in the x–y plane with its tail at the origin. Say it makes angle $\alpha$ with the x-axis and $\beta$ with the y-axis. Drop perpendiculars from A to each axis, meeting them at B and C. By the triangle rule,

We have split one arrow into two: one lying along x, one along y. The length OB is the projection of $\vec a$ on the x-axis, which is $a\cos\alpha$. Likewise OC $= a\cos\beta$.

Now bring in the unit vectors $\hat\imath$ and $\hat\jmath$, arrows of length 1 pointing along +x and +y. Then $\vec{OB} = a\cos\alpha\,\hat\imath$ and $\vec{OC} = a\cos\beta\,\hat\jmath$, so

Three dimensions#

A vector that leaves the x–y plane also casts a shadow on the z-axis. If it makes angles $\alpha, \beta, \gamma$ with the x, y and z axes, then with $\hat k$ the unit vector along z:

The numbers $a_x = a\cos\alpha$, $a_y = a\cos\beta$, $a_z = a\cos\gamma$ are the components of $\vec a$. Equation (2.3) says that every vector is a combination of the same three building blocks $\hat\imath, \hat\jmath, \hat k$.

More generally, the component of $\vec a$ along any direction that makes angle $\theta$ with it is $a\cos\theta$, which is its projection onto that direction. Nothing forces us to use x, y, z. Physicists often choose axes along a slope, along a rope, or along the motion, whichever makes the problem easiest.

Why components make addition easy#

Suppose we know three vectors by their components:

Adding them is just collecting like terms:

No triangles, no cosine rule, however many vectors you add. Try it with three arrows:

For vectors in the x–y plane the result is two perpendicular pieces, $R_x = a_x+b_x+c_x$ and $R_y = a_y+b_y+c_y$, so