Vectors in the Plane and Three dimensional Space
What is a Vector?
Vectors are often used in physics to represent quantities like force and velocity when we need to specify both the magnitude and direction of those quantities. Usually, we represent a vector visually by an arrow. In the applet below, vectors in and are shown.
Remark: There is a very special vector that does not have any direction. What is it?
Vectors can be drawn as arrows pointing out from the origin (the point where all axes intersect). On a coordinate space (either 2D or 3D), the coordinates of the point at the arrowhead uniquely determine the vector. Therefore, for the vectors in and in , we can express them mathematically as follows:
and
where and are points at the arrowhead of and i.e. P and Q respectively.
and are called the x-component and y-component of respectively.
Similarly, , , and are called the x-component, y-component, and z-component of respectively.
(You can freely drag the points P and Q on the arrowheads of both vectors in the applet above to change the length and the direction of the vectors.)
Moreover, two vectors are regarded as equal if they have the same length and direction. Hence, the tail of a vector can be any point other than the origin i.e. we may shift the vector to another position if necessary, as long as the length and direction remain unchanged. Therefore, whenever you are given a vector in or and want to express it mathematically, you can just shift the vector to the origin and find the coordinates of the arrowhead.
(You can freely drag the green vectors in the applet above to any position you like and they are considered as the same vectors as the black ones pointing out from the origin.)
Postion Vectors
As mentioned before, vectors are uniquely determined by the points at their arrowheads when pointing from the origin. Therefore, any point P (or Q) in (or ) can be represented by the vector with its tail at origin and its head at the point P (or Q). It is called the position vector of P (or Q). From this perspective, points and vectors are just two sides of the same coin. Sometimes it is convenient to regard points as position vectors.
Norm of a Vector
How can we compute the magnitude (length) of a vector? Thanks to Pythoragas Theorem, we have the following definitions:
Given in , the norm of the vector is
which is exactly the magnitude (length) of the vector .
For in , the norm of the vector is
which is exactly the magnitude (length) of the vector .
By definition, the norm of zero vector is zero and the norm of any non-zero vector is positive.
The following applet show how the above formulas are derived from Pythoragas theorem.
Vector Addition
There are two main operations on vectors. The first one is the addition of two vectors. First, we consider two vectors in the plane, we can define their addition visually using the applet below:
- Construct two vectors u and v in using the vector tool
. - Drag u to the origin i.e. the tail of u is a the origin.
- Drag v to the arrowhead of u.
- The vector u + v is defined as the vector pointing from the origin to the arrowhead of v.
Alternatively, you can regard the vector u + v as the "diagonal vector" of the parallelogram formed by the two vectors u and v pointing out from the origin. This is the so-called parallelogram rule. For the physics viewpoint, this definition of addition is quite natural. You can imagine two forces represented by u and v act on a mass at the origin. The resultant force is exactly u + v.
In the left pane of the above applet, when a vector u is created, its components are shown in the following format: . Observe the components of the vectors when you add two vectors together and answer the following question:
Question: Given vectors and , let . What are the components of ?
The addition of two vectors in are defined in the same way i.e. either using triangle rule or parallelogram rule. Moreover, suppose we are given two vectors and in ,
Question: What are the components of ?
Vector Scaling
The second main operation on vectors is scaling. Suppose k is any real number and u be any vector in or .
- If k >0, then ku is the vector having the same direction as u such that its length is k times the length of u.
- If k = 0, then ku is a zero vector.
- If k < 0, then ku is the vector having the opposite direction to u such that its length is |k| times the length of u. (Note: |k| is the absolute value of k.). In other words, .
Question: Given a real number k and vector in and vector in , what are the components of and ?
Unit Vectors
A vector is called a unit vector if its norm (length) equals 1. For any non-zero vector , we can scale it by factor to unit vector in the same direction:
Let . Then .
In , there are two special unit vectors and such that any vector in can be expressed in terms of and as follows:
.
Similarly, in , the three special unit vectors are , , and . Moreover, any vector in can be expressed as follows:
.
Vector Subtraction
Vector subtraction can be easily defined in terms of addition and scaling as follows: . Also, the components of can be computed by doing the subtraction component-wise.
You can construct vectors and in the above applet and then find out .
Question: Consider the parallelogram formed by two vectors and , can you express its two "diagonal vectors" in terms of and ?
Suppose and are two points in . The vector with as its tail and as its arrowhead is denoted by . As we know, and are the position vectors of and respectively. Therefore, we have
Since the norm of is exactly the distance between and , we have