1.9 Vector (cross) product  (Page 2/2)

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 :

$$\begin{array}{l}\mathbf{a}=a_x\mathbf{i}+a_y\mathbf{j}+a_z\mathbf{k}\\ \mathbf{b}=b_x\mathbf{i}+b_y\mathbf{j}+b_z\mathbf{k}\end{array}$$

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 :

Cross product of two vectors

$$\begin{array}{l}|\mathbf{a}\times \mathbf{b}|=ab\sin \theta \end{array}$$

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

$$\begin{array}{l}b\sin \theta =\mathrm{OB}\sin \theta =\mathrm{BD}\end{array}$$

Substituting, we have :

$$\begin{array}{l}|\mathbf{a}\times \mathbf{b}|=\mathrm{OA}x\mathrm{BD}=\mathrm{Base}x\mathrm{Height}=\mathrm{Area\; of\; parallelogram}\end{array}$$

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

$$\begin{array}{l}\mathbf{a}\times \mathbf{b}\ne \mathbf{b}\times \mathbf{a}\end{array}$$

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

$$\begin{array}{l}\mathbf{a}\times \mathbf{b}=-\mathbf{b}\times \mathbf{a}\end{array}$$

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 :

$$\begin{array}{l}\mathbf{a}\times (\mathbf{b}+\mathbf{c})=\mathbf{a}\times \mathbf{b}+\mathbf{a}\times \mathbf{c}\end{array}$$

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

$$\begin{array}{l}|\mathbf{a}\times \mathbf{b}|=ab\sin \theta \end{array}$$

or,

$$\begin{array}{l}|\mathbf{a}\times \mathbf{b}|=ax\left(b\sin \theta \right)\\ \Rightarrow |\mathbf{a}\times \mathbf{b}|=ax\mathrm{component\; of}\mathbf{b}\mathrm{in\; the\; direction\; perpendicular\; to\; vector}\mathbf{a}\end{array}$$

or,

$$\begin{array}{l}|\mathbf{a}\times \mathbf{b}|=bx\left(a\sin \theta \right)\\ \Rightarrow |\mathbf{b}\times \mathbf{a}|=ax\mathrm{component\; of}\mathbf{a}\mathrm{in\; the\; direction\; perpendicular\; to\; vector}\mathbf{b}\end{array}$$

4: Vector product in component form is :

$$\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_x& a_y& a_z\\ b_x& b_y& b_z\end{array}\right|$$

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 :

$$\begin{array}{l}\mathbf{a}\times \mathbf{b}=ab\sin \theta \mathbf{n}\end{array}$$

$$\begin{array}{l}\mathbf{a}\times \mathbf{b}=|\mathbf{a}\times \mathbf{b}|\mathbf{n}\end{array}$$

$$\begin{array}{l}\mathbf{n}=\frac{\mathbf{a}\times \mathbf{b}}{|\mathbf{a}\times \mathbf{b}|}\end{array}$$

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

$$\begin{array}{l}\mathbf{a}\times \mathbf{b}=ab\sin \theta \mathbf{n}=ab\sin 0°\mathbf{n}=0\end{array}$$

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

$$\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_x& a_y& a_z\\ b_x& b_y& b_z\end{array}\right|=0$$

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

$$\begin{array}{l}\frac{a_x}{b_x}=\frac{a_y}{b_y}=\frac{a_z}{b_z}\end{array}$$

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

$$\begin{array}{l}\mathbf{i}\times \mathbf{i}=\mathbf{j}\times \mathbf{j}=\mathbf{k}\times \mathbf{k}=0\end{array}$$

$$\begin{array}{l}\mathbf{i}\times \mathbf{j}=\mathbf{k};\mathbf{j}\times \mathbf{k}=\mathbf{i};\mathbf{k}\times \mathbf{i}=\mathbf{j}\\ \mathbf{j}\times \mathbf{i}=-\mathbf{k};\mathbf{k}\times \mathbf{j}=-\mathbf{i};\mathbf{i}\times \mathbf{k}=-\mathbf{j}\end{array}$$