Products of Vectors – Cheng Luyu

In this week’s fact of the week, I will introduce the vector dot and cross product, which may be already familiar to you. However, some of the mathematical perspective provided here may be useful in furthering your understanding in these operations. These operations are important in solving Olympiad physics problems and reading further in physics. These operations help to differentiate vectors from scalars in physics easily and solve the problem of judging the direction of certain vector, especially in three-dimensional space where intuition may fail.

The dot product is also called the scalar product. The result is a scalar quantity. In this article, all vectors are denoted by letters with arrow-heads, and their magnitudes by letters.

As defined, dot product —— (1),

where θ is the angle between and .

Alternatively, if and are both n dimensional vectors, i.e. they can be denoted in the form of , , then ——- (2).

Usually in physics, the vectors space will be two or three dimensional.

In the following part, I will provide a geometric perspective of dot product and derive (2) from (1) in two-dimensional space. The definition of dot product is subsequently generalised to n dimensional space.

Geometrically, the vector product is the product of magnitude of and the projection of on . Projection can be thought as a ‘shadow’. In the figure above, imagine if there exist parallel light beams perpendicular to shining towards , will leave a projection of magnitude on . The dot product is then the product of this scalar projection and the magnitude of itself. This is where the definition in (1) comes about.

A short digression here. In physics, this is a useful operation. For example, it avoids judging the sign of a scalar quantity obtained from calculation, like work. If we are only concerned with magnitude, . We will have to remember that s is the displacement in the direction of the force and if the two are in the opposite direction, W will be negative. However, avoids these considerations altogether. If the two are anti-parallel, from our definition, θ=180° and W will thus be the negative product of F and s (the magnitudes).

Now I will derive (2) from (1) in two dimensional plane. (I am not sure which definition arose first. Supposedly either one of the definitions can lead to the other. However, from what I read (1) came about first because the cross product was defined in two dimensional planes due to its usefulness in physics. Then with the correspondence between vectors and points in Cartesian plane, (2) is derived which is later generalised to n dimensional plane as a formal definition of scalar product in maths. You may want to find out more about the history, but anyway it is not our concern here.)

In the above diagram, let , . According to (1), .

Suppose OM direction is the positive x direction. Let and be the unit vector along positive x and y directions respectively.

Now we call , , , . Then , .

We know from addition formula that

—– (3)

Observe that

Sub in to (3), we have

Note that , , we have

Hence we have .

Note that this is applicable to three-dimensional (or higher) vector spaces.

Some properties are applicable to dot products:

(1)    Commutative: 

(2)    Distributive: 

I will now move on to cross product, which is also known as the vector product. It is denoted by . The result of cross product is a vector. Its direction is perpendicular to the plane which and lie in (note that any two intersecting lines determine a plane). The direction can be then specified by right-hand rule: curl the right hand from to . The direction in which the thumb points at is the direction of .

The magnitude is given by the formula , where θ is the angle between the two vectors.

I will avoid the tedious mathematical expressions that explain how to derive the result of a cross product here, but a useful expression can be used:

in matrix determinant form.

The way to calculate this ‘monster’ is to take the sum of product entries in the top-left-to-bottom-right diagonals minus the sum of product of entries in the top-right-to-bottom-left diagonals. If the diagonal is not complete (less than three entries), take the entries that are two skewed column from the column you are looking at.

If this looks abstract to you, consider the extended matrix as following:

Now the top-left-to-bottom-right diagonals and the top-right-to-bottom-left diagonals become pretty clear.

Then .

For those of you who are familiar with matrix, you can use the cofactor expansion along the first row as well.

Note that distributive rule still applies to cross product, but generally commutative property no longer holds. In fact, from definition, .

Vector product is very useful in electromagnetism. Where Fleming’s right-hand rule is a general rule of thumb, it may sometimes be confusing especially if the vectors are in three-dimensional space. Also, once is not orthogonal (perpendicular) to , it may be difficult to judge the direction. For example, in electromagnetism, if the field is not perpendicular to the length vector , it might not be so easy to figure out which angle to use or to determine the direction of the force. Writing equations in cross product provides a convenient way to avoid such confusion. In this case, by using , we can get the magnitude and the direction of force at one shot.

There are some interesting results related to operations of cross and dot products. I list some here. You may want to read more about their derivation or about more properties if you are interested.

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