The notion of vector

Vector calculus in plane and space: Vectors in Planes and Space

The notion of vector

In the plane and space we often work with vectors.

Vector

By vector we mean an arrow in the plane or space with a certain direction and length. In other words: a vector is a line segment which has a direction; the position of the vector in the plane or the space is not important. The norm or simply length of the vector is the length of the line segment.

When we drag a vector, so that its starting point is located elsewhere (but the direction and length remain unchanged), we consider this new arrow as a representative of the same vector.

Vectors are usually denoted by a character with a horizontal arrow above, which points to the right:\[\vec{v}\] The vector with a representative has a starting point #P# and an end point #Q# also indicated by #\vec{PQ}#.

There is one special vector, the vector of length \(0\). This vector does not provide a direction. We call this vector the null vector or zero vector and refer to it by \(\vec{0}\).

Position

A vector can be drawn in different places in the plane or in space.

Notation

In literature, we encounter varying notation of #\vec{v}#, such as \({\bf v}\), \(\underline{v}\), \(\bar{v}\), or just #v#.

Null vector

When #P# is a point of the space or the plane, the vector #\vec{PP}# is the null vector.

We now indicate how we can determine a vector by means of a point of the plane or space.

Correspondence between vector and point

In future, we will often encounter a chosen origin #O# in the plane or in the space. In situations such as this, we usually let the vector start at the origin. The vector is then solely determined by the endpoint, meaning that vectors can be seen as points of the space. The point #P# is then identified with the vector #\vec{OP}#. We then say that the vector #\vec{OP}# corresponds to the point #P#.

Confusion

We are not always consistent in the use of both perspectives (vector and point as a representative of the vector with starting point in the origin), but very often that does not lead to confusion or error.

LengthThe length of the vector #\vec{OP}# is the distance from #O# to #P#.

In coordinates The coordinates #\rv{x,y}# of a point of the plane determine the vector with starting point #\rv{0,0}# and endpoint #\rv{x,y}#. The vector in space with starting point #\rv{0,0,0}# and endpoint #\rv{x,y,z}# correspond to the point with coordinates #\rv{x,y,z}#.

We next indicate how the length of a vector in #\mathbb{R}^2# or #\mathbb{R}^3# can be expressed in terms of coordinates.

The length of a vector in coordinates

The length of a vector can be expressed in coordinates as follows.

The length of the vector #\rv{x,y}# in the plane #{\mathbb R}^2# is #\sqrt{x^2+y^2}#.
The length of the vector #\rv{x,y,z}# in the space #{\mathbb R}^3# is #\sqrt{x^2+y^2+z^2}#.

For #{\mathbb R}^2#, this result follows directly from the Pythagorean theorem. The length of the vector is, after all, the length of the hypotenuse of a right-angled triangle with sides of #|x|# and #|y|# in length.

Let #\rv{x,y,z}# be a point in #{\mathbb R}^3#. Then the length of the vector is equal to the length of the hypotenuse of a right-angled triangle with sides of #|z|# and #\sqrt{x^2+y^2}# in length (the length of the vector with coordinates #\rv{x,y}# in the horizontal plane). A second application of the Pythagorean theorem indicates that the length of that hypotenuse is equal to \[\sqrt{z^2+\left(\sqrt{x^2+y^2}\right)^2}=\sqrt{x^2+y^2+z^2}\]

Which of the following vectors shown in the plane below represent the vector #\rv{6,-0.7}#?

#t#, #u#, and #w# represent the vector, but #s# and #v# do not.

#s# does not have the right direction.

#v# does not have the right length.

If you drag the starting point from #t#, #u#, or #w# to #\rv{0,0}#, then the end point shows up at #\rv{6,-0.7}#.

New example