Direction of vector product

The two vectors (a and b) define an unique plane. The vector product is perpendicular to this plane defined by the vectors as shown in the figure below. The most important aspect of the direction of cross product is that it is independent of the angle, θ, enclosed by the vectors. The enclosed angle , θ, only impacts the magnitude of the cross product and not its direction.

Figure 1

Direction of vector product

Incidentally, the requirement for determining direction suits extremely well with rectangular coordinate system. We know that rectangular coordinate system comprises of three planes, which are at right angles to each other. It is, therefore, easier if we orient our coordinate system in such a manner that vectors lie in one of the three planes defined by the rectangular coordinate system. The cross product is, then, oriented in the direction of axis, perpendicular to the plane of vectors.

Figure 2

Direction of vector product

As a matter of fact, the direction of vector product is not yet actually determined. We can draw the vector product perpendicular to the plane on either of the two sides. For example, the product can be drawn either along the positive direction of y – axis or along the negative direction of y-axis (See Figure below).

Figure 3

Direction of vector product

The direction of the vector product, including which side of the plane, is determined by right hand rule for vector products. According to this rule, we place right fist such that the curl of the fist follows as we proceed from the first vector, a, to the second vector, b. The stretched thumb, then gives the direction of vector product.

Figure 4

Direction of vector product

When we apply this rule to the case discussed earlier, we find that the vector product is in the positive y – direction as shown below :

Figure 5

Direction of vector product

Here, we notice that we move in the anti-clockwise direction as we move from vector, a, to vector, b, while looking at the plane formed by the vectors. This fact can also be used to determine the direction of the vector product. If the direction of movement is anticlockwise, then the vector product is directed towards us; otherwise the vector product is directed away on the other side of the plane.

It is important to note that the direction of cross product can be on a particular side of the plane, depending upon whether we take the product from a to b or from b to a. This implies :

a × b ≠ b × a a × b ≠ b × a

Thus, vector product is not commutative like vector addition. It can be inferred from the discussion of direction that change of place of vectors in the sequence of cross product actually changes direction of the product such that :

Values of cross product

The value of vector product is maximum for the maximum value of sinθ. Now, the maximum value of sine is sin 90° = 1. For this value, the vector product evaluates to the product of the magnitude of two vectors multiplied. Thus maximum value of cross product is :

The vector product evaluates to zero for θ = 0° and 180° as sine of these angles are zero. These results have important implication for unit vectors. The cross product of same unit vector evaluates to 0.

The cross products of combination of different unit vectors evaluate as :

There is a simple rule to determine the sign of the cross product. We write the unit vectors in sequence i,j,k. Now, we can form pair of vectors as we move from left to right like i x j, j x k and right to left at the end like k x i in cyclic manner. The cross products of these pairs result in the remaining unit vector with positive sign. Cross products of other pairs result in the remaining unit vector with negative sign.

Cross product in component form

Two vectors in component forms are written as :

a = a x i + a y j + a z k b = b x i + b y j + b z k a = a x i + a y j + a z k b = b x i + b y j + b z k

In evaluating the product, we make use of the fact that multiplication of the same unit vectors gives the value of 0, while multiplication of two different unit vectors result in remaining vector with appropriate sign. Finally, the vector product evaluates to vector terms :

Evidently, it is difficult to remember above expression. If we know to expand determinant, then we can write above expression in determinant form, which is easy to remember.

Geometric meaning vector product

In order to interpret the geometric meaning of the cross product, let us draw two vectors by the sides of a parallelogram as shown in the figure. Now, the magnitude of cross product is given by :

Figure 6: Two vectors are represented by two sides of a parallelogram.

Cross product of two vectors

| a × b | = a b sin θ | a × b | = a b sin θ

We drop a perpendicular BD from B on the base line OA as shown in the figure. From ΔOAB,

b sin θ = OB sin θ = BD b sin θ = OB sin θ = BD

Substituting, we have :

| a × b | = OA x BD = Base x Height = Area of parallelogram | a × b | = OA x BD = Base x Height = Area of parallelogram

It means that the magnitude of cross product is equal to the area of parallelogram formed by the two vectors. Thus,

Since area of the triangle OAB is half of the area of the parallelogram, the area of the triangle formed by two vectors is :

Attributes of vector (cross) product

In this section, we summarize the properties of cross product :

1: Vector (cross) product is not commutative

a × b ≠ b × a a × b ≠ b × a

A change of sequence of vectors results in the change of direction of the product (vector) :

a × b = - b × a a × b = - b × a

The inequality resulting from change in the order of sequence, denotes “anti-commutative” nature of vector product as against scalar product, which is commutative.

Further, we can extend the sequence to more than two vectors in the case of cross product. This means that vector expressions like a x b x c is valid. Ofcourse, the order of vectors in sequence will impact the ultimate product.

2: Distributive property of cross product :

a × ( b + c ) = a × b + a × c a × ( b + c ) = a × b + a × c

3: The magnitude of cross product of two vectors can be obtained in either of the following manner :

| a × b | = a b sin θ | a × b | = a b sin θ

or,

| a × b | = a x ( b sin θ ) ⇒ | a × b | = a x component of b in the direction perpendicular to vector a | a × b | = a x ( b sin θ ) ⇒ | a × b | = a x component of b in the direction perpendicular to vector a

or,

| a × b | = b x ( a sin θ ) ⇒ | b × a | = a x component of a in the direction perpendicular to vector b | a × b | = b x ( a sin θ ) ⇒ | b × a | = a x component of a in the direction perpendicular to vector b

4: Vector product in component form is :

a × b = | i j k a x a y a z b x b y b z | a × b = | i j k a x a y a z b x b y b z |

5: Unit vector in the direction of cross product

Let “n” be the unit vector in the direction of cross product. Then, cross product of two vectors is given by :

a × b = a b sin θ n a × b = a b sin θ n

a × b = | a × b | n a × b = | a × b | n

n = a × b | a × b | n = a × b | a × b |

6: The condition of two parallel vectors in terms of cross product is given by :

a × b = a b sin θ n = a b sin 0 ° n = 0 a × b = a b sin θ n = a b sin 0 ° n = 0

If the vectors involved are expressed in component form, then we can write the above condition as :

a × b = | i j k a x a y a z b x b y b z | = 0 a × b = | i j k a x a y a z b x b y b z | = 0

Equivalently, this condition can be also said in terms of the ratio of components of two vectors in mutually perpendicular directions :

a x b x = a y b y = a z b z a x b x = a y b y = a z b z

7: Properties of cross product with respect to unit vectors along the axes of rectangular coordinate system are :

i × i = j × j = k × k = 0 i × i = j × j = k × k = 0

i × j = k ; j × k = i ; k × i = j j × i = - k ; k × j = - i ; i × k = - j i × j = k ; j × k = i ; k × i = j j × i = - k ; k × j = - i ; i × k = - j