I decided to take on MIT Multivariable Calculus to get a review for next semester. Having some struggles with this question though and the solutions aren't really helping me out much, thinking I might be seeing this in a naíve way and am missing a step on how they got there.
Suppose a plane $ax+by+cz = d$
We are supposed to prove the formula $D = \frac{\mid d \mid}{\sqrt{a²+b²+c²}}$
where D is the distance to the origin.
I was having some troubles with the proof, checked the solution and slowly started to get a clue, but need to see if I'm getting this right.
So we suppose a $P_0 = (x_0,y_0,z_0)$ and know that the normal vector is $\vec{N} = (a,b,c)$.
The solution started like this $\vec{OP}.\frac{\vec{N}}{|\vec{N}|}$ and then by usual operations it got to the formula that we wanted.
So my question is, am I supposed to see the dot product as the projection of the vector that connects the origin to a certain point into the direction of the Normal vector? I'm getting that value that will be the minimal distance?
The intuition is to see the dot product as some kind of parser that finds the commonalities between 2 vectors and returns the vector that is composed by those commonalities?
Also finally, I understand that dividing by the length of the Normal vector gives us the "unitary" direction vector right? But I'm not seeing the implication of not doing that, how would it deform the resulting distance? Would it make larger or smaller depending on the length of the normal vector that we choose?
If these questions are basic I'm sorry, I never had much love for this class and hopefully will get better in these next few weeks as I watch this course.
Best Answer
I think you have it pretty much right. The dot product can be understood as a measure of alignment between two vectors. For example, if $\vec{u}$ and $\vec{v}$ are two unit vectors, then $$\vec{u}\cdot\vec{v} = \cos\theta$$ where $\theta$ is the angle between the two vectors. So in this case $\vec{u}\cdot\vec{v} = 1$ means the vectors point in the same direction (i.e. $\vec{u}=\vec{v}$), $\vec{u}\cdot\vec{v} = 0$ means the vectors are perpendicular, and $\vec{u}\cdot\vec{v} = -1$ means the vectors point in opposite directions (i.e. $\vec{u}=-\vec{v}$.)
In the more general case in which the vectors are not unit vectors, we have $$\vec{u}\cdot\vec{v} = |\vec{u}| |\vec{v}| \cos\theta$$ which means that $\vec{u}\cdot\vec{v}$ takes on its largest possible value, $|\vec{u}| |\vec{v}|$, when the vectors point in the same direction.
In the case where one of the vectors (say $\vec{v}$) is a unit vector, $\vec{u} \cdot \vec{v}$ measures the component of $\vec{u}$ in the $\vec{v}$ direction. If we introduce the notation $\hat{v}$ to stand for $\frac{\vec{v}}{|\vec{v}|}$, then $\hat{v}$ is a unit vector in the $\vec{v}$ direction, and we have that the component of $\vec{u}$ in the $\vec{v}$ direction is
$$\operatorname{comp}_{\vec{v}}\vec{u} = \vec{u} \cdot \hat{v} = \frac{\vec{u}\cdot\vec{v}}{|\vec{v}|}$$
It's important to realize that $\operatorname{comp}_\vec{v} \vec{u}$ is a scalar quantity, measuring the "size" of the component of $\vec{u}$ in the direction of $\vec{v}$. (I put "size" in quotes because this scalar also includes a sign, so it's not just a magnitude.) It's also useful to define a vector $\operatorname{proj}_\vec{v} \vec{u}$ as follows:
$$\operatorname{proj}_\vec{v} \vec{u} = \left( \operatorname{comp}_\vec{v} \vec{u} \right) \hat{v}$$
Notice that $\operatorname{proj}_\vec{v} \vec{u}$ points in (or opposite to) the $\vec{u}$ diction and has a length equal to $|\operatorname{comp}_\vec{v} \vec{u}|$. Unpacking this definition, we have the following equivalent forms:
$$\operatorname{proj}_\vec{v} \vec{u} = \left( \vec{u} \cdot \hat{v}\right) \hat{v}$$ $$\operatorname{proj}_\vec{v} \vec{u} = \left( \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|}\right) \frac{\vec{v}}{|\vec{v}|}$$ $$\operatorname{proj}_\vec{v} \vec{u} = \left( \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2}\right) \vec{v}$$