Class 11 > Unit # 02: Kinematics > Cross Product or Vector Product

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Unit # 2: Kinematics — Cross Product / Vector Product and Characteristics

Lecture Overview

This lecture explores the definition, direction determination via right-hand rule, mathematical formulation, and key characteristics of the Vector or Cross Product of two vectors.

Source Video: Cross Product / Vector Product , Characteristics - Unit 2 Kinematics - Class 11 Physics

1. Definition & Physical Example of Cross Product

When two vectors are multiplied together in such a way that the resulting product is a vector quantity (having both magnitude and a specific direction), the operation is called a vector product or cross product. It is denoted by placing a cross (\(\times\)) between the two vectors.

Physical Example:

• Torque or Moment of Force (\(\vec{\tau}\)): Defined as the cross product of position vector (\(\vec{r}\)) and Force (\(\vec{F}\)):
  $$\vec{\tau} = \vec{r} \times \vec{F}$$

2. Mathematical Expression & Direction

Consider two vectors, \(\vec{a}\) and \(\vec{b}\), inclined at an angle \(\theta\) in a plane. The cross product results in a new vector \(\vec{c}\):

$$\vec{c} = \vec{a} \times \vec{b} = ab \sin\theta \, \hat{n}$$

Where:

• \(ab \sin\theta\) represents the magnitude of the vector product.
• \(\hat{n}\) is a unit vector perpendicular to the plane containing both \(\vec{a}\) and \(\vec{b}\), pointing in the direction given by the right-hand rule.

3. Key Characteristics of Cross Product

I. Non-Commutative Law

The cross product does not obey the commutative law. Reversing the order reverses the direction of the resulting vector:

$$\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$$

II. Perpendicular Vectors

If two vectors are perpendicular (\(\theta = 90^\circ\)), their cross product magnitude is maximum because \(\sin(90^\circ) = 1\):

$$\vec{a} \times \vec{b} = ab \, \hat{n}$$

III. Parallel Vectors

If two vectors are parallel (\(\theta = 0^\circ\)), their cross product is zero because \(\sin(0^\circ) = 0\):

$$\vec{a} \times \vec{b} = 0$$

IV. Anti-Parallel Vectors

If two vectors are anti-parallel (\(\theta = 180^\circ\)), their cross product is also zero because \(\sin(180^\circ) = 0\):

$$\vec{a} \times \vec{b} = 0$$

V. Self Cross-Product of a Vector

The cross product of a vector with itself is always a zero vector:

$$\vec{a} \times \vec{a} = 0$$

VI. Cross Product of Unit Vectors

For orthogonal unit vectors \(\hat{i}, \hat{j}, \hat{k}\):

• Self-units (Zero): \(\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0\)
• Cyclic order: \(\hat{i} \times \hat{j} = \hat{k}\), \(\quad \hat{j} \times \hat{k} = \hat{i}\), \(\quad \hat{k} \times \hat{i} = \hat{j}\)
• Anti-cyclic order: \(\hat{j} \times \hat{i} = -\hat{k}\), \(\quad \hat{k} \times \hat{j} = -\hat{i}\), \(\quad \hat{i} \times \hat{k} = -\hat{j}\)

VII. Rectangular Components Form (Determinant Form)

Given vectors in component form \(\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}\) and \(\vec{b} = b_x\hat{i} + b_y\hat{j} + b_z\hat{k}\), the cross product is evaluated using a determinant:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$$

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