Calc3.2

Vectors, and Vector Functions

Vectors

A vector in the plane.

A vector is a mathematical object which has magnitude like a number, but also direction. For example, if I wanted to describe how a person is driving at any particular time, I would need two numbers: their speed, and some measure of their direction (the angle it makes with true north, for example). These two quantities together constitute the motion vector of that person.

We also use vectors to describe position: for example, to locate an object, it suffices to know how far it is from you, and in what direction.

In three dimensions, three component numbers are required to describe a vector.

Vectors are the bread and butter of multivariable calculus because they pack multiple variables into a single, easily manipulated quantity. Let's look at some more specific examples.

In our first descriptions, we used a coordinate system you should be familiar with call "polar coordinates." If you aren't familiar with this system, we will review them shortly. When using Cartesian coordinates, vectors are very, very familiar. The point (x,y) has a position vector, or put more simply, is the vector written Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}x\\y\end{pmatrix} \ } . See illustration at right. Unlike variables which may be numbers, like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, x, \ } etc., vectors are usually written ${\vec {w}}\$ or $\mathbf {w} \$. While the latter is common in print and online, we will use the former since it is more easily duplicated in handwriting.

In three dimensional space we use three variables to write down a vector. In Cartesian coordinates, this would look like ${\vec {w}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}\$.

We can, if we wish, talk about just the length of the vector - this is its absolute value, or $\left\|{\vec {w}}\right\|\$. For a vector written in Cartesian coordinates, the Pythagorean theorem tells us this is just ${\sqrt {x^{2}+y^{2}}}\$.

Vectors can be added and subtracted, just like numbers. See the illustration below.

Vectors are added by connecting them head to tail. Vectors are subtracted by reversing heads and tails for the subtracted vector, and then connecting them head to tail.

In Cartesian coordinates, this is particularly easy: ${\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}x+a\\y+b\end{pmatrix}}\$ and ${\begin{pmatrix}x\\y\end{pmatrix}}-{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}x-a\\y-b\end{pmatrix}}\$.

Multiplication by a number is also simple: the vectors direction is unchanged, but its length is multiplied by that number. Again, in Cartesian coordinates, this becomes simple: $c{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}ca\\cb\end{pmatrix}}\$.

The dot product is a measure of the length of one vector in the direction of another.

There are two ways to multiply a vector by another vector. In two dimensions, we can perform only one, which is called the dot product or in more advanced terminology, inner product. Given two vectors, this dot product is how far one vector goes in the direction of the other, using the other as the "unit" of length. It is defined ${\vec {u}}\cdot {\vec {w}}=\left\|{\vec {u}}\right\|\left\|{\vec {w}}\right\|\cos(\theta )\$, where $\theta \$ is the angle between ${\vec {u}}\$ and ${\vec {w}}\$. As before, this takes a particularly simple form in Cartesian coordinates: ${\begin{pmatrix}x\\y\end{pmatrix}}\cdot {\begin{pmatrix}a\\b\end{pmatrix}}=xa+by\$. This expression is similar for the three dimensional case: ${\begin{pmatrix}x\\y\\z\end{pmatrix}}\cdot {\begin{pmatrix}a\\b\\c\end{pmatrix}}=xa+yb+zc\$.

The cross product of two vectors.

While the dot product can be used in any number of dimensions, there is a special way of multiplying vectors which only exists in three dimensions (although it is a specific case of a more general concept). It is called the cross product and written ${\vec {u}}\times {\vec {w}}\$. The cross product is defined as having length equal to the area of the parallelogram borderd on two sides by ${\vec {u}}\$ and ${\vec {w}}\$, and direction perpendicular to either vector, with a right handed orientation. This means that if we take our right hand, point the palm in the direction of ${\vec {u}}\$, bend our knuckles so that they point in the direction of ${\vec {w}}\$, then our thumb is already pointing in the direction of ${\vec {u}}\times {\vec {w}}\$. This definition is not easy to calculate with, and shortly we will introduce a more convenient way of calculating these kinds of products.

While we often write ${\vec {w}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}\$ to describe a vector, we often find it more convenient to define three vectors ${\vec {i}}={\begin{pmatrix}1\\0\\0\end{pmatrix}},{\vec {j}}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\$ and ${\vec {k}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}\$, unit vectors which point along the positive $x,y,\$ and $z\$ axes, respectively.

With these vectors, we can write our original vector ${\vec {w}}=x{\vec {i}}+y{\vec {j}}+z{\vec {k}}\$. It means exactly the same thing as ${\begin{pmatrix}x\\y\\z\end{pmatrix}}\$, but is a little friendlier to the printed page. As an additional notational convenience, we often write ${\vec {w}}=w_{1}{\vec {i}}+w_{2}{\vec {j}}+w_{3}{\vec {k}}\$, since keeping track of all the $x\$'s and $a\$'s would get unwieldy very fast if we are working with several vectors.

As we are about to see, these conveniences can make the calculation of the cross product relatively simple. Hopefully, the student will remember the definition of the determinant of a matrix, because this will make calculating cross products much, much easier. For those who are unfamiliar with the idea, we will review the concept.

For a matrix ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}\$, we define the determinant ${\begin{vmatrix}a&b\\c&d\end{vmatrix}}$ to be $ad-bc\$. To compute the determinant of a larger square matrix, then, we choose any row of the matrix and take the left entry, and multiply that number by the determinant of the matrix formed by eliminating the first row and column of the original matrix. From this, we subtract the next entry in our row times the matrix formed by eliminating THAT row and column of the orginal matrix. Then we add the next term, and so on and so forth until we have a complete expansion.

Below is an example of this technique, expanding along the first row.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{vmatrix}a&b&c\\r&s&t\\x&y&z\end{vmatrix} = a\begin{vmatrix}s&t\\y&z\end{vmatrix} - b \begin{vmatrix}r&t\\x&z\end{vmatrix} + c \begin{vmatrix}r&s\\x&y\end{vmatrix}}

Examine this equation, and make sure you understand how each of the matrices on the right hand side were formed. Once we have reached this point, we can use our definition of a 2x2 determinant to arrive at a value.

Now, we finally arrive at a means of calculating the cross product of two vectors. It turns out that the definition we gave of cross product amounts to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}\times\vec{w}=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\u_1&u_2&u_3\\w_1&w_2&w_3\end{vmatrix} \ }

Normally, we don't allow vectors (like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{i} \ } ) to be elements of a matrix or a determinant, but in this case, we make an exception because the formula is so useful and easy to remember. We will encounter a similar exception to the rules when we give an easy way to remember the formula for curl.

Vector Functions

Vector functions are often visualized as fields, with the output vectors drawn with their base at the point that generated that output.

Before, we described scalar functions as taking a point in space and ascribing a number to it - for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y,z) = x^2 + y^2 +z^2 \ } . With our new notation, we can describe this as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\vec{r}) = \vec{r} \cdot \vec{r} \ } , which is a scalar function written with vectors.

There is another class of functions, though, which return not numbers but vectors. These are vector functions, and they are useful for describing many phenomenon, most notably flows, winds, and heat dispersion. For example, at every point in the ocean, the water is moving with both a speed and a direction. If we have some vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r} \ } which can be used to desrcibe points in the ocean, we can write a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\vec{r}) = \vec{v} \ } , which will give us the velocity vector of the water at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r} \ } . Very often, we use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r} \ } as a variable vector describing a position or location, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} \ } for velocity, and later we'll encounter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a} \ } for acceleration.

Coordinate Systems

The spherical coordinate system.

We're all very familiar with the Cartesian coordinates in two dimensions, and once the student has learned the "right hand rule" and committed it to memory, they have learned all there is about the Cartesian system in three dimensions.

Hopefully, the students will be familiar with polar coordinates as well, but we shall review.

In the plane, it is frequently useful to describe shapes, not by their distance along two axes, but by their straight-line distance to a point, and their angle to a fixed line. We use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r,\theta) \ } instead of the usual Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y \ } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \ } is the distance and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \ } is the angle.

We can easily move between Cartesian and polar coordinates with these equations:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \sqrt{x^2+y^2} \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \arctan{\frac{y}{x}} \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=r \arccos{(\theta)} \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=r \arcsin{(\theta)} \ }

Cylindrical coordinates are just polar coordinates, with a z-axis. They are frequently useful for describing phenomenon which have circular motion in a plane, but are moving in some direction (like a spiral).

More common than cylindrical coordinates are spherical coordinates, which use distance from a center point together with "latitude and longitude" to describe points in three dimensional space.

In the spherical coordinate system, a point is described with its distance to a central point, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \ } , the angle made in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy \ } plane between the points projection into that plane and the x-axis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \ } , and the angle made between the point and the z-axis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \ } .

As with polar coordinates, there are equations which allow us to transfer between spherical and Cartesian coordinates.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = \sqrt{x^2 + y^2 + z^2} \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \arctan{(\frac{y}{x})} \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi = \arctan{ \left({ \frac{\sqrt{x^2+y^2}}{z} }\right) } \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = r \sin(\phi) \cos(\theta) \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = r \sin(\phi) \sin(\theta) \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = r \cos(\phi) \ }

Limits of Multivariate Functions

The limit of a multivariate function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{(x,y)\rightarrow(x_0,y_0)} f(x,y) \ } is almost identical to the single-variable limit in concept, but complicated by the realization that (x,y) can approach the target point from any direction, not just the left or right hand sides.