Multivariable Lesson 1.1: Vectors and Space Curves

Hopefully you are already familiar with vectors. A vector is, in brief, a magnitude and a direction. This is in contrast to a scalar, which is only a magnitude. Vectors are usually represented graphically by arrows of length proportional to their magnitude, and pointing in their direction. A vector doesn’t specify where it is situated, only how it points, so you can slide a vector around anywhere. This is an important point

Vectors can exist in any number of dimensions, but we will be primarily concerned with three-dimensional ones. These vectors are usually best manipulated in (rectangular) component form. A vector (1, 2, 3) can be thought of as 1i+2j+3k, where i, j, and k are the “basis vectors”, unit vectors which point in the directions of the positive x-, y-, and z-axes, respectively. That is, they are equal to (1, 0, 0), (0, 1, 0), and (0, 0, 1), respectively.

The Dot Product

Because they may (read: will) come in handy later, we will now define the dot and cross product operations between two vectors. The dot product between two vectors u=(u1, u2, u3) and v=(v1, v2, v3) is denoted u dot v and is a scalar equal to u1v1+u2v2+u3v3. It extends similarly to any dimension. The dot product is useful in many different ways: for example, dotting a vector with itself results in the square of its magnitude. (Convince yourself of this.) Most importantly, perhaps, is that u dot v = |u| |v| cos theta, where |u| denotes the magnitude of u, and theta is the angle between the two vectors. By extension, since the cosine of pi/2 is 0, the dot product of two orthogonal (perpendicular) vectors is zero.

This last equation can be used to derive the formulas for vector projections. Projecting vector u onto v is the process by which u is represented as the sum of two perpendicular vectors: one which is parallel to v and one which is perpendicular to v. The magnitude of the parallel vector is called the scalar projection of u onto v and is denoted comp_v u. The actual parallel vector itself is called the vector projection, and is denoted proj_v u.

Diagram of the scalar projection

How can we use dot products for this? Well, if we look at the diagram, we see that we can form a right triangle by dropping a perpendicular from u onto v. The scalar projection is simply the length of the leg on v, or |u| cos theta. We can rewrite this more conveniently using our formula from above: comp_v u=|u| cos theta = |u||v|cos theta / |v| = u dot v/|v| To find the vector projection, we need to multiply this, the magnitude of the projection, by a unit vector pointing in the direction of v: proj_v u = (u dot v/|v|)(v/|v|) = (u dot v/|v|^2)v

The Cross Product

In three dimensions, there are infinitely many directions orthogonal to one vector, but only two (opposite) directions in which a vector may be perpendicular to two nonparallel vectors at once. We now invent an operation called the cross product, which takes two nonparallel 3D vectors and returns a vector orthogonal to both. u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1) Happily, there is a mnemonic for this operation if you recall how to take determinants: u x v is the determinant of the matrix [[i, j, k][u1, u2, u3][v1, v2, v3]] While not a critical point, it is perhaps worth noting that this is an abuse of notation: vectors cannot really be elements of a matrix. Anyway, several questions are immediately raised. For example, what is the magnitude of the resulting vector? It turns out that |u x v| = |u| |v| sin theta And because sin 0 = 0, the cross product of parallel vectors is always zero.

Diagram of the right hand rule

One last ambiguity that must be discussed is which of the two directions the cross product will return. To determine which direction u x v points, we use the “right hand rule”. Using your right hand, point your index finger in the direction of u, point your middle finger in the direction of v, and (assuming you feel no discomfort) your thumb will be pointing in the direction of u x v.

Space Curves

Graph of r(t)=(cos t, sin t, 3t)

Vectors don't have to be composed of constants, of course. To represent a vector which varies with a parameter (let’s say time), we can create a function which takes a scalar parameter as input and outputs a vector rather than the scalar output to which we’re accustomed. We’ll call this a vector function. For example, r(t)=(cos t, sin t, 3t) would be a vector function. (Note: by convention, r(t) always represents a vector function.)

If we consider that this vector’s components indicate a location in coordinate space by always placing the tail of the vector at the origin, then we can graph this function as we’d graph any parametric function: by varying t and graphing all the values of the function. The result will be a three-dimensional helix, an example of what we’ll call a space curve. The graph of the function we just defined is shown to the right.

Not surprisingly, we will be able to use calculus to analyze these space curves.

The right hand rule graphic was adapted from ExploreLearning’s Magnetic Induction Gizmo.

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