Can someone explain to me what this means? I understand the part that says the scalar projection of vector $u$ onto vector $v$ is $|u|\cos(\theta)$.
But I don't understand what the vector projection means. Why is the dot product of $v$ and $u$ divided by the magnitude of $v$ squared? Why does that give us the vector projection?
Thomas Calculus Early Transcendentals 12th txtbk.pdf (page 713 of 1211)
Figure 12.25 The length of $\text{proj}_v\mathbf u$is (a) $|\mathbf u|\cos\theta$ if $\cos\theta\ge0$ and (b) $-|\mathbf u|\cos\theta$ if $\cos\theta<0$
The number $|\mathbf u|\cos\theta$ is called the scalar component of $\mathbf u$ in the direction of $\mathbf v$ (or of $\mathbf u$ onto $\mathbf v$). To summarize,
The vector projection of $\mathbf u$ onto $\mathbf v$ is the vector $$\text{proj}_v\mathbf u=\left(\frac{\mathbf u\cdot\mathbf v}{|\mathbf v|^2}\right)\mathbf v\tag1$$
The scalar component of $\mathbf u$ in the direction of $\mathbf v$ is the scalar $$|\mathbf u|\cos\theta=\frac{\mathbf u\cdot\mathbf v}{|\mathbf v|}=\mathbf u\cdot\frac{\mathbf v}{|\mathbf v|}\tag2$$
Note that both the vector projection of $\mathbf u$ onto $\mathbf v$ and the scalar component of $\mathbf u$ onto $\mathbf v$ depend only on the direction of the vector $\mathbf v$ and not its length (because we dot $\mathbf v$ with $\mathbf v/|\mathbf v|$, which is the direction of $\mathbf v$).
Best Answer
It has two parts:
The direction where you're projecting onto. That's the unit vector direction of $v$, which is computed by dividing $v$ by the length of $v$. That is $\dfrac{v}{|v|}$.
The component of $u$ in the direction of $v$. That is, the "shadow" or image of $u$ when you project it onto $v$. This is computed by $\dfrac{u \cdot v}{|v|}$. Why? Let's revisit the definition of the dot product: $u\cdot v = |u||v|\cos(\theta)$. Hence $|u| \cos(\theta) = \dfrac{u \cdot v}{|v|}$ and that gives you (as in the triangle figure in your book), the length of $u$'s projection on the direction of $v$.
Putting it together, the projection of $u$ onto $v$ is a vector of length $\dfrac{u \cdot v}{|v|}$ in the direction of $\dfrac{v}{|v|}$, i.e. $$ \dfrac{u \cdot v}{|v|}\dfrac{v}{|v|} = \dfrac{u \cdot v}{|v|^2}v $$