Chapter 2: Vectors review You have already been exposed to the geometric fundamentals of vectors and their use in various calculations; specifically in terms of a vector having a magnitude and direction. Defining vectors in this oversimplified manner is useful for a broad range of applications but this definition breaks down in advanced work1. Some exploration of the properties of vectors will be useful in showing some of the unique and powerful features vectors have.
Vector definition: Let us define a vector as a linear independent combination of a set of components within an associated basis. For example, an arbitrary vector, , is expressed as:
The components , and are specified by the choice of coordinate system (CS) as indicated by the basis unit vectors , and . For nearly every case in this course, we will assume an orthogonal rectangular CS where, and represent the directions east-west (x-direction), north-south (y-direction) and upward-downward (z-direction) respectively. The magnitude of vector is defined as
The unit direction can then be found for the vector by dividing (1) by (2)
Let us consider a more simple, 2-D vector in the - plane, . The components of can be decomposed geometrically by the use of trigonometric identities for a right triangle
The relationships of equations (3) can be easily deduced from Figure 1.
Figure 1 – Right triangle showing the relationship between the vector, , and its horizontal and vertical components.
Vector notation There are a few ways to express vectors.
I. Arrow () - A vector is identified with an arrow on top of the variable of interest.
II. Boldface (B) - A vector is identified by the use of bold-face font.
III. Underline () - A vector is identified by the use of underlining the function.
Vector properties: The section covers a list of the algebraic properties of vectors:
1. Two vectors can added together to produce a new vector:Vector addition is defined by summing the associated components in a given CS. As an example, for vectors and the addition of these vectors leads to a new vector defined as:
Equating the components of to the sum of and , we can see that
; ; (5)
Geometrically the property of vector addition is seen by adding placing the “tail” of the vector to the “head” of the vector . The resultant vector from the tail of vector to the head of the vector is the new vector . A 2-D example of this geometric relationship is shown in Figure 2.
Figure 2 – Addition of two vectors and , resulting in
For completeness, it is stated without proof that vector addition is both associative and commutative:
2. A scalar quantity can be multiplied with a vector to produce a vector: Given a vector and a scalar quantity, scalar multiplication of on leads to a new vector, , which is defined by distributing the scalar quantity to each component of :
Equation (6) shows us that the components of are
Geometrically, scalar multiplication on a vector causes a stretching or compression to the length of the vector. If the scalar quantity, , then the resultant vector is stretched as compared to the original vector. Alternatively, when , the new vector is compressed. If , then the resultant vector is in the opposite direction of the original vector. Using the scalar with the property of vector addition, we can also subtract vectors. A 2-D example of scalar multiplication, for ,and , is shown in Figure 3.
Figure 3 – Scalar multiplication on the vector for the scalar quantities,and
3. Scalar (Dot) product: The scalar or dot product is a way to multiply two vectors together such that the result is a scalar. For vectors and the dot product is algebraically defined as
Using equation (7) it can be shown from simple trigonometry that the dot product can also be defined “geometrically” as
Where is the co-planar angle between vectors and .
Question: How would you go about use equation (7) to show equation (8)?
Equation (8) shows us that the dot product is a measure of the “parallelness” of two vectors. When the vectors and are parallel, and the dot product has maximum value. Alternatively, if is perpendicular to , then and so .
We will often indicate two vectors are perpendicular to each other by showing that their dot product is 0. It is important for you to have clear understanding of this fact.
Example – Given the following vectors, find the coplanar angle between them:
4. Cross product: The vector cross product is a way to multiply two vectors together such that the result is a vector. For vectors and the cross product is algebraically defined as
A convenient way to express the cross product is in terms of the determinant:
As was the case for the dot product, we can obtain a “geometric” definition of the cross product from equation (9). The result is,
Where is the co-planar angle between vectors and and is a unit vector (a vector with a magnitude of 1) normal to the co-plane formed by vectors and .
Equation (11) shows us that the cross-product is a measure of how perpendicular two vectors are. When is perpendicular to , then and the cross product is a maximum. Alternatively, if is parallel to , then and .2
We will often indicate two vectors are parallel to each other by showing that their cross product is 0. Example: The geostrophic wind equation is expressed as where is the velocity vector, and and are vectors that we will learn about later in this course.
Answer provided:Taking the dot product of on both sides of , we obtain:
Now we know from equation (11) that the resultant vector, , produced by is perpendicular to both and . In other words, is perpendicular to and . Therefore using equation (8), we can see that .
Example: Occasionally we must find the triple cross product of two vectors. For example, for a general position vector and Coriolis vector, the following cross product may come up
In such cases, the following identity is helpful
- Use the identity in equation (12) to express the as a linear combination of the vectors and
Answer: Using equation (12) we can see that
can be expressed as
Revisiting the concept of a vector as a magnitude and direction: As stated earlier, the general vector in equation (1) is specified by a set of components whose values are determined by a specific CS. Physically, the choice of CS is simply the point of view of the observer of the problem. We expect that the laws of nature do not depend on the location or perspective of the observer and therefore we expect that the physical properties of vectors will not depend on the point of view of the observer. In other words, vectors must contain some property that is invariant for any CS. It turns out that the invariant properties of the vector that are of interest to us are the magnitude and relative direction of two vectors from each other.
Consider the following example:
A two-dimensional vector is represented in Figure 4 below.
Figure 4. Diagram showing a vector
Now, Figure 5 shows the same vector in a CS that is rotated in the horizontal plane about the axis by 10 degrees. With a good eye and a steady ruler, one can show that the vector has the form in this rotated CS.
Figure 5. Vector expressed in a CS rotated 10 degrees CCW.
Although there is a new set of components and basis for, it is still the same vector. To confirm that the vector is the same let us first calculate the magnitude of in both CSs, by use of equation (2).
For the first CS, and for the rotated CS, . Thus we can see the magnitude of is the same in both CSs even though the orientation of the vector is different with respect to the axis of each CS.
It is easy to see that computing the magnitude of a vector is related to taking the scalar product or Dot Product. Specifically, we are taking the square root of the scalar product of the vector with itself. Recall from equation 10 that the dot product of two vectors in an orthogonal Cartesian CS with basis vectors , and :
is expressed as
Note there are no basis vectors in the above expression, indicating that the dot product produces a scalar value. If we now perform a rotation of the coordinates system, we will observe that the dot product of the vectors remains the same even though both vectors have new coordinates.
We prove this result as follows:
For two vectors, , , we have already seen that the vector magnitude is invariant to CS transformations and so the scalar quantitiesand are also invariant to CS transformations since these quantities are just the square of the magnitude. Further the magnitude of the vector is also invariant and so must . Using the foil method on this new vector we see that
Solving for in the above we see that
Now all the quantities on the right side are simply vector magnitudes and therefore must be invariant to CS transformations. The left hand side therefore must also be invariant to CS transformation. Thus we have proved that the scalar product does not change under CS transformation.
Given that the dot product does not change under coordinate transformation, we can deduce that the angle between two vectors does not change either. Recall the geometric definition of the dot product:
Where is the co-planar angle between vector and . Given the invariance of the dot product and the magnitude of each vector to coordinate rotations, we also know that the angle between the two vectors is also invariant under coordinate rotation.
In summary, we have observed three significant properties of vectors:
1. The magnitude of a vector is invariant under coordinate rotation.
2. The dot product of two vectors is invariant under coordinate rotation.
3. The angle between two vectors is invariant under coordinate rotation.
These three properties allow us to state a useful result for vectors:
Vectors are objects in space which have a magnitude and a relative direction between each other. Both of these vector characteristics are invariant to any change in the coordinate system. You might ask yourself how this can this result be of any use to us? The key is that, when solving problems, we can choose any CS we wish without changing the physics. If we have a vector in an orthogonal CS like the left hand side of figure 6, we can rotate the CS such that the vector is aligned along the x-axis of the new CS. Then we can express the vector simply as .
Figure 6 – A vector expressed in two different CS. In the left CS the vector has the form . In the right CS the vector is expressed as .
1 For specific examples, see Arfken and Weber, 1995. Mathematical Methods for Physicists 4th ed., Academic Press, San Diego. Page 8.
2 Notice that this result is a zero vector and not just the number zero; the difference being that which is a vector which has 0 for all its components. It is common, however, to leave the vector symbol off of . In other words, it is common to see the result since there really is no way to specify an actual direction.