What is the dot product of two or three vectors graphically or visually?

Question

I don't understand what the dot product actually is. I understand when and where to use it, but when it comes to proving things with it, I don't really grasp what it actually is making it difficult to comprehend.

Answer 1

You're asking two question: what is the dot product and what it means visually.

Your first question is fairly straightforwards: Given vectors $\vec v=(v_1,v_2,\ldots, v_n)$ and $\vec w=(w_1,w_2,\ldots,w_n)$ in $\mathbb R^n$, $\vec v\cdot\vec w=v_1w_1+v_2w_2+\ldots+v_nw_n$. Just multiply the corresponding components and add up all the products. For example $(2,3,5)\cdot(1,4,5)=2+12+25=39$.

Your second question doesn't have a precise answer; it's hard to really see the dot product because it is a scalar and not a vector (or more precisely, a function that takes two vectors and returns a scalar), but it's still possible to get a sense of its geometric meaning.

It might be useful to recall the identity $\vec v \cdot \vec w=||\vec v||\cdot ||\vec w|| \cos\theta$, where $\theta$ is the angle between $\vec v$ and $\vec w$.

This tells you that the dot product is a way of measuring how much two vectors point in the same direction, in a way that takes magnitude into account.

Loosely and informally, it answers the question "how long is the agreement between these two vectors?" (This question is of course too imprecise to be anything more than intuition.)

Here are some facts that illustrate this point of view: Given two vectors of fixed length, their dot product will be as big as possible (and equal to the the product of their lengths) if they point in the same direction, and as small as possible (a negative number of the same size as above) if they point in opposite directions.

If the vectors are perpendicular, ie. pointing neither towards nor away from eachother, then their dot product is 0.

Furthermore, when $\vec u$ is a unit vector, $\vec v \cdot \vec u$ tells you how much $\vec v$ points in the direction of $\vec u$; that is, $\vec v \cdot \vec u$ is the length of the projection of $\vec v$ onto $\vec u$.

Answer 2

You cannot dot three vectors, only two. This is because when you dot two vectors you get a scalar. The dot product of two vectors divided by the magnitudes of the two vectors gives the cosine of the angle between the vectors.

Answer 3

The dot product, or scalar product, takes a pair of vectors (in any dimension) and gives a number, i.e. a scalar. (They both have to be from the same dimension.) If you have two vectors, say \begin{eqnarray*} {\bf u} &=& (u_1,u_2,\ldots,u_n) \\ {\bf v} &=& (v_1,v_2,\ldots,v_n) \end{eqnarray*} Then the dot product of ${\bf u}$ and ${\bf v}$, denoted by ${\bf u}\cdot{\bf v}$ or $\langle {\bf u},{\bf v}\rangle$, is given by the number $${\bf u}\cdot {\bf v} = u_1v_1 + u_2v_2 + \cdots +u_nv_n$$ The dot product has two geometrical interpretations. They come from the formula $$\cos\theta = \frac{{\bf u}\cdot {\bf v}}{\|{\bf u}\| \|{\bf v}\|}$$ where $\theta$ is the angle between ${\bf u}$ and ${\bf v}$. Assuming that ${\bf u}$ and ${\bf v}$ are non-zero, then ${\bf u}\cdot {\bf v} = 0$ if, and only if, $\cos \theta =0$. In other words, for non-zero ${\bf u}$ and ${\bf v}$, we have ${\bf u}\cdot {\bf v} = 0 \iff {\bf u} \perp {\bf v}$. (The symbol $\perp$ means "perpendicular to", i.e. "at a right angle to".)

If ${\bf u} = {\bf v}$ then $\theta =0$ and $\cos\theta =1$. Hence $$\frac{{\bf u}\cdot {\bf u}}{\|{\bf u}\| \|{\bf u}\|}=1 \implies {\bf u}\cdot {\bf u} = \|{\bf u}\|^2$$ The dot product allows us to measure angles between vectors and the lengths of vectors. The geometrical meaning of the dot product is that it gives angles and lengths.