Geometry

Vectors, points and arrows

A 3×1 column-vector or 3-vector (x y z)^T determines a point (x, y, z) in the three-dimensional Cartesian space. Conversely, every point in this space is determined by a vector.

For any point (a, b, c) in space, the vector (x y z)^T also determines an arrow (or directed line segment, if you like) with tail at that point and head at the point (a + x, b + y, c + z). (The book refers to such an arrow as a vector, but I shall avoid doing so.)

One purpose of allowing the tail-point of an arrow to vary is this. Suppose x is the vector (x₁ x₂ x₃)^T, and y is the vector (y₁ y₂ y₃)^T. Then x determines the arrow from the origin (0, 0, 0) to (x₁, x₂, x₃), and y determines the arrow from this point to (x₁ + y₁, x₂ + y₂, x₃ + y₃), and this is the point determined by x + y.

Norm

The norm of a 3-vector is the length of the arrow from the origin to the point that the vector determines. If u is the vector (u₁ u₂ u₃)^T, then the norm of u is the non-negative scalar |u| such that

|u|² = u₁² + u₂² + u₃² .

(Note that the norm is often indicated by double bars, rather than single as here.) In particular, if k is a scalar, then

|ku| = |k||u| ,

that is, the norm of ku is the product of the absolute value of k and the norm of u.

Dot-product

Definition

Suppose also v is the vector (v₁ v₂ v₃)^T. Then each of u and v determines an arrow with tail-point at the origin; let θ (Greek small letter theta) be the angle from the one arrow to the other (assuming the arrows have positive length). With the origin, the head-points determine a triangle; we have

|u - v|² = |u|² + |v|² - 2|u||v| cos θ

by the Law of Cosines. The dot-product of u and v is the scalar

|u||v| cos θ

appearing in the equation; it is denoted by u·v. The Law of Cosines yields a formula:

u·v = u₁v₁ + u₂v₂ + u₃v₃ ;

this can be taken as an alternative definition of the dot-product, and it makes sense regardless of the norms of the vectors.

If the norms of u and v are positive, then angle θ is acute, right or obtuse, respectively, if and only if u·v is positive, zero or negative. Regardless of the norms, we shall say that two vectors are orthogonal if their dot-product is zero. [The symbol for orthogonality is an upside-down T.]

Properties

The dot-product is an example of a scalar product of vectors; we shall see others later. The dot-product is like a product of scalars in the following ways:

• It is symmetric: u·v = v·u.
• It distributes over addition: u·(v + w) = u·v + u·w.
• It associates with scalar multiplication: k(u·v) = ku·v.
• v·v ≥ 0 always, and v·v = 0 if and only if v = 0.

The last property holds because |u|² = u·u.

Projections

Suppose a is a non-zero vector. There is a vector ka such that u - ka is orthogonal to a (and hence to ka). In fact, we can calculate that

k = (u·a)/(a·a).

We call ka the projection of u on a and denote it by proj_au. We may say that proj_au is parallel to a, since it is a scalar multiple of a. So then u is the sum of two orthogonal vectors, namely proj_au and u - proj_au, the first being parallel to a.

Cross-product

The appropriate changes being made, everything till now in this section makes sense in the Cartesian plane. The same is not true for the cross-product, which is defined only in 3-space.

Theoretical definition

If u and v are 3-vectors, then there is a 3-vector w such that

det (u v x) = w·x

for any 3-vector x. The vector w is called the cross-product of u and v, denoted by u×v.

Practical definition

You can check that the previous definition justifies the following method for calculating the cross-product. Suppose i, j and k are the vectors determining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1) respectively. You can write down what looks like a 3×3 matrix, with u as the first column, and v as the second column, but where the entries of the third column are i, j and k respectively. If you calculate the determinant formally, you get a linear combination of i, j and k, which is a vector itself: it is the cross-product u×v.

Properties

The cross-product is anti-symmetric: v×u = -u×v. From the theoretical definition, it follows that u×v is orthogonal to both u and v. Using the facts about dot-products, you can also show

|u×v| = |u||v|sin θ.

This means that the norm of the cross-product of u and v is the area of the parallelogram determined by u and v (that is, the parallelogram whose vertices are the origin and the points determined by u, v and u + v).

If there is a third 3-vector, x, then the three vectors u, v and x determine a parallelepiped. [The word is Greek: parallel + epi + ped[on]; an epipedon is a surface; compare parallel + o + gram[mon], where gramma is a line.] The next equation is a restatement of the theoretical definition of the cross-product:

(u×v)·x = det (u v x).

The volume of the parallelepiped is the absolute value of either side of this equation.

In 2-space, two vectors determine a parallelogram, whose area is the absolute value of the determinant of the matrix whose columns are the two vectors.

Next section: n-dimensional space