Scalar and vector projections

Given two vectors, can we ask how much of one vector is pointing in the direction of the other? We can certainly ask how much of the vector $\left<5,6\right>$ is pointing in the $x$ direction – the answer is just 5. You can think of this as projecting the vector onto the $x$-axis and asking for its projected length. Similarly we can ask about the projection of a vector into any arbitrary direction. This is illustrated in figure \ref{vec6}. Imagine having a light perpendicular to $\vec{b}$ shining towards it. There is a shadow of the vector $\vec{a}$ cast on the line of $\vec{b}$. This is the scalar projection of $\vec{a}$ in the direction of $\vec{b}$, also called the component of $\vec{a}$ in the direction of $\vec{b}$. When you are looking at this, clearly the size of $\vec{b}$ is unimportant, so you can think of an infinite line stretching in both directions parallel to $\vec{b}$.

We can form the vector $\vec{a}$ as the addition of the blue vector, and the red vector. The blue vector is the component of $\vec{a}$ which is parallel to $\vec{b}$ and the red vector is the component of $\vec{a}$ which is perpendicular to $\vec{b}$. Because the length of the vector $\vec{b}$ is unimportant to find the component of $\vec{a}$ in the direction of $\vec{b}$ we can simply think of the direction of $\vec{b}$ as being defined by the infinite line (shown in light grey) as the extension of $\vec{b}$ in both directions.

To calculate this quantity, which we call $comp_{\vec{b}}\vec{a}$ it is clearly just:

$comp_{\vec{b}}\vec{a}=\left|\vec{a}\right|\cos\theta=\frac{\vec{a}.\vec{b}}{\left|\vec{b}\right|}$

Check to make sure that you understand why the last equality is so. We can see that this therefore has nothing to do with the length of $\vec{b}$ as we stated before, just its direction.

Here we have just asked how much of $\vec{a}$ is pointing in the direction of $\vec{b}$. We can also define a vector of this length, in the direction of $\vec{b}$. This is simply the quantity we already have multiplied by a unit vector in the direction of $\vec{b}$ and thus given by:

$proj_{\vec{b}}\vec{a}=\frac{\vec{a}.\vec{b}}{\left|\vec{b}\right|}\frac{\vec{b}}{\left|\vec{b}\right|}=\frac{\vec{a}.\vec{b}}{\left|\vec{b}\right|^2}\vec{b}$

Hint: Make sure that you can derive these quantities (the comp and proj) yourself. Don’t just remember the formulae but understand where they come from!

 How clear is this post?