David Pierce | Matematik | M.S.G.S.Ü.

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Vectors, points and arrows

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

For any point (abc) 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 + xb + yc + 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 (x1x2x3), and y determines the arrow from this point to (x1 + y1x2 + y2x3 + y3), and this is the point determined by x + y.


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.



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.]


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:

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


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 projau. We may say that projau is parallel to a, since it is a scalar multiple of a. So then u is the sum of two orthogonal vectors, namely projau and u - projau, the first being parallel to a.


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.


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×vx = 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

Son değişiklik: Friday, 15 January 2016, 16:06:46 EET