Projections of Vectors

• Projection of $\vec{u}$ on $\vec{v}$ is kinda like how much of $\vec{u}$ would lie on $\vec{v}$.
• Projections are like the $x,y$ components of our original vector.

$$\begin{array}{r}{\text{proj}}_{\vec{v}}(\vec{u})=\frac{\vec{v}\cdot \vec{u}}{||\vec{v}|{|}^2}\vec{v}\\ =(\vec{v}\cdot \vec{u})\left(\frac{\vec{v}}{||\vec{v}||}\right)\frac{1}{||\vec{v}||}\end{array}$$

• From physics, we know that the horizontal component (the projection) is $\vec{r}\cos \theta$.
• Here we are just expressing it vectorially, without involving angles.
• The dot product helps us replace the angle, but there are two problems with it.
• First, dot product is scaler. It does not automatically tell us which direction the resulting projection points toward. We can use intuition in problems, but we also need a concrete math definition we can fall back on.
• Second, while the dot product helps us replace the angle, it also brings along the second vector with it.
• We solve the direction issue by calculating the unit vector of the projected-upon vector. Giving us the first fraction-bracket.
• The second fraction helps us undo the dot product effect (partly). We are basically cancelling out the second vector.

Perp

• Perp is the vertical component of the projection.
• Since everything here is a vector:

$${\text{perp}}_{\vec{v}}(\vec{u})=\vec{u}-{\text{proj}}_{\vec{v}}(\vec{u})$$

• Both are perpendicular to each other.

Directions of Projections

• Use the cosine, angle rules from dot product to figure it out.