# Dot Product Of Vector Functions

The problem I am working on is:

Find $\vec{r}(t) \cdot \vec{u}(t)$ Is the result a vector-valued function?

I was able to find the dot product, it was quite easy. What I am having difficulty with is understanding what sort of information does the dot product of two vectors reveal about the two vector valued functions.

I know that $\vec{r}(t)$ is a vector valued function that is a function of the parameter $t$, where $t-$values (the inputs) yield a vector, whose initial point is the origin, and terminal point is a point on the graph that the $t-$value determines from the coordinate functions. Is this a correct interpretation?

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Before we discuss vector-valued functions, it is important to interpret the dot-product between two fixed vectors. Most important, the dot product between two vectors is a scalar (typically a real or complex number). Geometrically, for vectors $u,v$ in Euclidean space, the dot product obeys the general formula $$u \cdot v = \|u\| \|v\| \cos{\theta}$$ where $\theta$ is the angle between $u$ and $v$, and $\| \cdot \|$ indicates the length of the vector. For two vectors lying on a plane, it is a bit easier to visualize. Notice that if $\theta = \pi/2$, then the dot product is $0$, so orthogonal vectors yield a zero dot product. Furthermore, if $\theta = 0$, then then dot product is as large as possible if we hold the length of the vectors constant. What this means is that the size of the dot product combines two pieces of information:

1. The lengths of the vectors involved.
2. How close to parallel the vectors are.

If we just know the dot product, we can't separate these two; that is, if the dot product is big, it could be that the vectors are long, or that the vectors are very close to parallel, or both, whereas if the dot product is small, it could be that the vectors are short, or that they are very close to perpendicular, or both.

Now, turning to vector-valued functions, $r(t)$ and $u(t)$ each give you a vector at each particular "time" $t$, and so the function $r(t) \cdot u(t)$ is a scalar function that tells you the dot product at each given $t$; this product precisely measures the relationship between $r(t)$ and $u(t)$ as described above.

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If the dot denotes the inner product, then it is scalar valued.

The zeroes of this function are the values $t$ where the vectors $r(t)$ and $u(t)$ are orthogonal.

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