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Source: Wikipedia

We know from basic trigonometry, the projection of vector \(\vec{A}\) onto vector \(\vec{B}\) (see figure to the right) is just \( \|\vec{A}\| \cos(\theta)\). However, what if we don't know the value of angle \(\theta\)? We can use the dot product to find the projection of one vector onto another. Let's derive the equation.

From the picture on the right, it is easy to see that the magnitude of the projection of vector \(\vec{A}\) onto vector \(\vec{B}\) is \( \|proj_{\vec{B}} \vec{A} \| = \|\vec{A}\| \cos(\theta)\). Since we don't know the angle \(\theta\), we can use the angle between vectors formula to substitute for \(\cos(\theta)\) as follows.

\(\begin{array}{rcl} \| proj_{\vec{B}} \vec{A}\| & = & \|\vec{A}\| \cos(\theta) \\ & = & \displaystyle{ \|\vec{A}\| \frac{\vec{A} \cdot \vec{B}}{\|\vec{A}\| \|\vec{B}\|} } \\ & = & \displaystyle{ \frac{\vec{A} \cdot \vec{B}}{\|\vec{B}\|} } \end{array}\)

Notice that this result is just the magnitude of the projection of vector \(\vec{A}\) onto \(\vec{B}\). If we want to find the projection vector itself, we can multiply this result by the unit vector in the direction of \(\vec{B}\) and end up with \[ proj_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{\|\vec{B}\|} \frac{\vec{B}}{\|\vec{B}\|} = \frac{\vec{A} \cdot \vec{B}}{\|\vec{B}\|^2} \vec{B} \] This last equation is the preferred equation since calculating \(\|\vec{B}\|\) requires taking a square root, so squaring it removes that difficulty.

Here is a great video explaining vector projection in more detail.

Now that we have the projection vector, it is a simple matter to find the vector component of \(\vec{A}\) orthogonal to \(\vec{B}\) (the dashed line in the above picture) using basic addition of vectors. Let's call our unknown vector \(\vec{v}\). We know that \( proj_{\vec{B}} \vec{A} + \vec{v} = \vec{A}\). Solving for \(\vec{v}\), we have the vector component of \(\vec{A}\) that is orthogonal to \(\vec{B}\) which is \[ \vec{v} = \vec{A} - proj_{\vec{B}} \vec{A} \]

We highly recommend that you spend some going through this section carefully until you understand it well. When you get to lines and planes, you will need to understand and be able to use this concept.

You may find different notation for the projection vector than we used here. The notation \( proj_{\vec{B}} \vec{A} \) is fairly common but not used exclusively.