Cross Product

The cross product is a mathematical operation in vector algebra. It takes two vectors in three-dimensional space and produces a new vector that is perpendicular to both. Unlike the dot product, which gives a scalar, the cross product results in a vector. Its magnitude represents the area of the parallelogram formed by the original vectors. Its direction is determined by the right-hand rule. This makes it an important tool in physics, engineering, and computer graphics. It is used to calculate torque, rotational direction, and surface normals. By combining direction and magnitude, the cross product offers a strong way to analyse and solve problems related to spatial relationships.

Table of Contents

• What is the Cross Product?
• Cross Product Formula
• Right-hand Rule Cross Product
• Properties of the Cross Product
• Cross Product Example in 3D
• Misconceptions
• Solved Examples
• Conclusion
• FAQs on Cross Product

 

What is the Cross Product?

The cross product, also known as the vector product, is a binary operation that takes two vectors in three-dimensional space and produces a new vector. 

• This new vector is perpendicular to both original vectors. 

• Its magnitude equals the area of the parallelogram formed by the two vectors. 

• Its direction is determined using the right-hand rule. 

This property makes the cross product a useful tool in physics, rotational motion, torque calculation, and 3D geometry.

 

Cross Product Formula

The cross product is a way to multiply two vectors in 3D space. It produces a new vector that is perpendicular to both original vectors. The magnitude of the cross product represents the area of the parallelogram formed by the two vectors.

• Geometric Form
  A→ × B→ = |A→| |B→| sinθ n̂
  (where n̂ is a unit vector perpendicular to A→ and B→)

 

• Component Form
  If A→ = (a, b, c) and B→ = (x, y, z), then:
  A→ × B→ = (bz – cy)i→ + (cx – az)j→ + (ay – bx)k→

 

• Determinant Form
  A→ × B→ =

| i→ j→ k→ |
| a b c |
| x y z |

 

Example
Let X→ = 5i→ + 6j→ + 2k→ and Y→ = i→ + j→ + k→.

Step 1: Write determinant form:
X→ × Y→ =

| i→ j→ k→ |
| 5 6 2 |
| 1 1 1 |

Step 2: Expand:
= i→(6×1 – 2×1) – j→(5×1 – 2×1) + k→(5×1 – 6×1)

Step 3: Simplify:
= 4i→ – 3j→ – k→

Final Answer:
X→ × Y→ = 4i→ – 3j→ – k→

Right-hand Rule Cross Product

The cross product of two vectors gives a vector that is perpendicular to both of them. The direction of this new vector is found using the Right-hand Rule.

What is the Right-hand Rule?

• Stretch out your right hand.

• Point your index finger in the direction of the first vector (A).

• Point your middle finger in the direction of the second vector (B).
  
  
• Your thumb now points in the direction of the cross product (A × B).
  
  

Formula:
If A→ = (a₁, a₂, a₃) and B→ = (b₁, b₂, b₃), then
A→ × B→ = (a₂b₃ – a₃b₂)i→ – (a₁b₃ – a₃b₁)j→ + (a₁b₂ – a₂b₁)k→

Magnitude:
|A→ × B→| = |A→||B→|sinθ

Example:
Let A→ = (2, 3, 4) and B→ = (5, 6, 7)

A→ × B→
= (3×7 – 4×6)i→ – (2×7 – 4×5)j→ + (2×6 – 3×5)k→
= (21 – 24)i→ – (14 – 20)j→ + (12 – 15)k→
= –3i→ + 6j→ – 3k→

Final Answer:
A→ × B→ = (–3, 6, –3)

 

The result is perpendicular to both A and B, and its direction is decided by the Right-hand Rule.

 

Properties of the Cross Product

Anti-Commutative Property

The cross product changes sign if you swap the order of the vectors.
Formula: A→ × B→ = −(B→ × A→)
Example: Let A→ = (1, 0, 0) and B→ = (0, 1, 0).
A→ × B→ = (0, 0, 1) and B→ × A→ = (0, 0, −1).

Therefore, A→ × B→ = −(B→ × A→).

 

Distributive Property

The cross product distributes over vector addition.
Formula: A→ × (B→ + C→) = A→ × B→ + A→ × C→
Example: Let A→ = (1, 0, 0), B→ = (0, 1, 0), and C→ = (0, 0, 1).
B→ + C→ = (0, 1, 1).
A→ × (B→ + C→) = (0, −1, 1).
A→ × B→ + A→ × C→ = (0, 0, 1) + (0, −1, 0) = (0, −1, 1).

 

Scalar Multiplication Property

A scalar can be factored out in the cross product.
Formula: (kA→) × B→ = k(A→ × B→) = A→ × (kB→)
Example: Let k = 2, A→ = (1, 0, 0), and B→ = (0, 1, 0).
(2A→) × B→ = (2, 0, 0) × (0, 1, 0) = (0, 0, 2).
A→ × B→ = (0, 0, 1), then 2(A→ × B→) = (0, 0, 2).

 

Zero Vector Property

If two vectors are parallel or one is the zero vector, the cross product is the zero vector.
Formula: A→ × B→ = 0→
Example: A→ = (1, 2, 3) and B→ = (2, 4, 6) (B→ is parallel to A→).
A→ × B→ = (0, 0, 0).

 

Magnitude Property

The magnitude of A→ × B→ equals the area of the parallelogram they form.
Formula: |A→ × B→| = |A→||B→|sin(θ)
Example: A→ = (3, 0, 0), B→ = (0, 4, 0), and θ = 90°.
|A→ × B→| = (3)(4)sin(90°) = 12, so the area = 12 units².

 

Cross Product Example in 3D

Given:
A→ = 2i→ + 3j→ + k→
B→ = i→ − 4j→ + 5k→

 

Using the determinant formula:

A→ × B→ =

| i→  j→  k→ |
| 2  3  1 |
| 1 −4  5 |

= i→(3×5 − 1×−4) − j→(2×5 − 1×1) + k→(2×−4 − 3×1)
= i→(15 + 4) − j→(10 − 1) + k→(−8 − 3)
= 19i→ − 9j→ − 11k→

 

Final Answer:
A→ × B→ = 19i→ − 9j→ − 11k→

 

Magnitude:
|A→ × B→| = √(19² + (−9)² + (−11)²)
= √(361 + 81 + 121)
= √563

 

Area Interpretation: 
The magnitude √563 represents the area of the parallelogram formed by vectors A and B in 3D space.

.

Misconceptions

• The cross product constantly affects a scalar.

Reality: The product of vectors is a vector, no longer a scalar. It has both magnitude and route.

• The go product is commutative.

Reality: The cross product is anti-commutative, that means A × B = - (B × A). Changing the order reverses the route of the result.

• The cross product is defined for any size.

Reality: The goal product is simply properly defined in a three-dimensional area. It no longer generalises clearly to other dimensions.

• A × A isn't always zero.

Reality: Any vector crossed with itself offers the 0 vector, since the angle between them is zero° and sin(zero) = 0.

• Cross product measures similarity.

Reality: The pass product measures perpendicularity, no longer similarity. It is used to discover a vector orthogonal to two others.

 

Solved Examples

Example 1: Simple Cross Product

Problem: A→ = i→ + j→, B→ = j→ + k→. Find A→ × B→.
Given: A→ = (1, 1, 0), B→ = (0, 1, 1)

Steps:
Use the formula:
A→ × B→ = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)

Compute components:
x = 1·1 − 0·1 = 1
y = 0·0 − 1·1 = −1
z = 1·1 − 1·0 = 1

So A→ × B→ = (1, −1, 1)
Final Answer: A→ × B→ = i→ − j→ + k→

 

Example 2: Cross Product is Zero (parallel vectors)

Problem: A→ = 2i→, B→ = 4i→. Find A→ × B→.
Given: A→ = (2, 0, 0), B→ = (4, 0, 0)

Steps:
x = 0·0 − 0·0 = 0
y = 0·4 − 2·0 = 0
z = 2·0 − 0·4 = 0

So A→ × B→ = (0, 0, 0)
Final Answer: A→ × B→ = 0→

 

Example 3: Right-Hand Rule (direction)

Problem: If A→ points east and B→ points north, what is the direction of A→ × B→?

Steps:
east = +x, north = +y
x × y = +z (by right-hand rule)

Final Answer: A→ × B→ points upward (out of the plane), in the +z direction

 

Example 4: Cross Product and Area

Problem: A→ = 3i→, B→ = 4j→. Find |A→ × B→| (area of the parallelogram).
Given: A→ = (3, 0, 0), B→ = (0, 4, 0)

Steps:
x = 0·0 − 0·4 = 0
y = 0·0 − 3·0 = 0
z = 3·4 − 0·0 = 12

So A→ × B→ = (0, 0, 12)
|A→ × B→| = √(0² + 0² + 12²) = 12

Final Answer: Area = |A→ × B→| = 12

 

Example 5: Torque (r × F)

Problem: r→ = 2i→ + 3j→, F→ = 4j→ + k→. Find torque T→ = r→ × F→.
Given: r→ = (2, 3, 0), F→ = (0, 4, 1)

Steps:
Determinant form:
T→ = | i→ j→ k→ |
   | 2 3 0 |
   | 0 4 1 |

x = 3·1 − 0·4 = 3
y = 0·0 − 2·1 = −2
z = 2·4 − 3·0 = 8

So T→ = (3, −2, 8)
Final Answer: T→ = 3i→ − 2j→ + 8k→

 

Conclusion

 The cross product is a basic vector operation that creates a vector perpendicular to two given vectors in three-dimensional space. This idea is commonly used in physics, engineering, and 3D geometry, especially for analysing rotational motion, torque, and directional vectors. It has defining properties like anti-commutativity and the assurance of producing a perpendicular vector. The cross product improves spatial visualisation and problem-solving abilities in vector mathematics.

 

Frequently Asked Questions on Cross Product

1. What does the cross product mean? 

Answer: It gives you a vector that is at a right angle to two other vectors in 3D space.

 

2. What’s the cross product of i and j? 

Answer: When you cross i and j, you get k, which is a unit vector pointing up along the z-axis.

 

3. What about the cross product of A and B? 

Answer: A crossed with B gives you a vector that’s perpendicular to both, and its size is based on the lengths of A and B and the angle between them.

 

4. What if two cross products are equal? 

Answer: If A crossed with B equals C crossed with D, it means both sets of vectors create the same perpendicular vector in both direction and size.

 

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