**Vector Product of Two Vectors a and b is:
**The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. It is denoted by x (cross).

A x B = AB sin θ \(\hat{\mathbf{n}}\)

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## Vector or Cross Product of Two Vectors in Physics – Scalars and Vectors

**Vector Cross Product Properties**

(i) Vector product is not commutative, i.e.

**A** x **B** ≠ **B** x **A** [∴ (**A** x **B**) = – (**B** x **A**)]

(ii) Vector product is distributive, i.e.

**A** x (**B** + **C**)= **A** x **B** + **A** x **C
**

(iii) Vector product of two parallel vectors is zero, i.e.

**A**x

**B**=

*AB*sin 0° = 0

(iv) Vector product of any vector with itself is zero.

**A** x **A** = *AA* sin 0° = 0

(v) Vector product of orthogonal unit vectors

\(\hat{\mathbf{i}} \times \hat{\mathbf{i}}=\hat{\mathbf{j}} \times \hat{\mathbf{j}}=\hat{\mathbf{k}} \times \hat{\mathbf{k}}=0\)

and \(\hat{\mathbf{i}} \times \hat{\mathbf{j}}=-\hat{\mathbf{j}} \times \hat{\mathbf{i}}=\hat{\mathbf{k}}\)

\(\hat{\mathbf{j}} \times \hat{\mathbf{k}}=-\hat{\mathbf{k}} \times \hat{\mathbf{j}}=\hat{\mathbf{i}}\)

\(\hat{\mathbf{k}} \times \hat{\mathbf{i}}=-\hat{\mathbf{i}} \times \hat{\mathbf{k}}=\hat{\mathbf{j}}\)

(vi) Vector product in cartesian coordinates

\(\mathbf{A} \times \mathbf{B}=\left(A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}+A_{z} \hat{\mathbf{k}}\right) \times\left(B_{x} \hat{\mathbf{i}}+B_{y} \hat{\mathbf{j}}+B_{z} \hat{\mathbf{k}}\right)\)

= (A_{y}B_{y} – A_{z}B_{y})\(\hat{\mathbf{i}}\) – (A_{x}B_{z} – B_{x}A_{z})\(\hat{\mathbf{j}}\) + (A_{x}B_{y} – A_{y}B_{x})\(\hat{\mathbf{k}}\)

**Direction of Vector Cross Product**

When **C** = **A** x **B**, the direction of **C** is at right angles to the plane containing the vectors **A** and **B**. The direction is determined by the right hand screw rule and right hand thumb rule.

(i) **Right Hand Screw Rule:
**Rotate a right handed screw from first vector

**(A)**towards second vector

**(B)**. The direction in which the right handed screw moves gives the direction of vector

**(C)**.

(ii) **Right Hand Thumb Rule:
**Curl the fingers of your right hand from A to B. Then, the direction of the erect thumb will point in the direction of

**A**x

**B**.

**Scalars and Vectors Topics:**