Multiplication of Vectors

A vector in mathematics is defined by its magnitude and direction, representing both the size and orientation of a quantity in space.

For example, a force vector’s magnitude shows how strong the force is, while its direction shows where it is applied. In the case of displacement, the magnitude tells how far something moves, and the direction shows the path taken. 

Vectors have various notational forms depending on the dimensionality of their space.
In two-dimensional (2D) space, a vector, often denoted as v, can be represented as a column matrix with two components:

\(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix} \)

Similarly, in three-dimensional (3D) space, a vector has three components and is written as:

 \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix} \)

  
Alternatively, vectors can be expressed in terms of their components, often using unit vectors (like i, j, k for 3D) or simply listed with commas. For example, \(3i + 2j + k \)

Vectors inherently possess both magnitude and direction; the idea of "multiplication" expands beyond the single operation we know for regular numbers (scalars). These two fundamental properties necessitate different types of vector multiplication, each yielding a unique result tailored for specific applications.


Dot Product: The dot product is also known as the scalar product, where two vectors are combined, and it results in a scalar product. It is the product of the magnitudes of the vectors and the cosine of the angle between them. The dot product tells us how much one vector extends in the direction of the other, and the result can be positive or negative.


For example, the dot product of two non-zero vectors a and b with an angle θ between them can be written as: 

\(a · b = |a| |b| cos(θ)\)

Where |a| and |b| are the magnitudes of vectors a and b
θ is the angle between them.

This operation is useful for determining the relationship between vectors, such as finding the angle between them or checking if they are perpendicular (in which case their dot product results in zero when the vectors are neither a zero vector)

The notation for the dot product of vectors is \(a \cdot b \).


The Cross Product: The cross product is also known as the vector product. Here, two vectors are multiplied together to result in a third vector. The third vector is perpendicular to both original vectors, and it is determined by the right-hand rule. The magnitude of the new vector is based on the magnitudes of the original vectors and the sine of the angle between them (representing the area of the parallelogram they form.)

The cross product is applicable in 3D space, for two vectors in the X-Y plane, their cross product yields a vector along the Z-axis, which is perpendicular to the vectors. 
It is represented by \( \mathbf{a} \times \mathbf{b} = |\mathbf{a}|\,|\mathbf{b}| \sin(\theta)\,\hat{n} \)

Where is the angle between them
\(\hat{n}\) is the unit vector. 
 

The methods for multiplying vectors, including both the dot product and the cross product, are explained in the following sentences.
 

Dot Product: The dot product of two vectors in component form using the unit vectors i, j, and k is calculated by multiplying the corresponding components and adding the results. The unit vectors \( \hat{i}, \; \hat{j}, \; \hat{k} \) are mutually perpendicular, and their magnitude is 1. 

For example, finding the dot product of vectors a and b, 

\( \vec{a} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k} \) and \( \vec{b} = a_2 \hat{i} + b_2 \hat{j} + c_2 \hat{k} \)

\( \vec{a} \cdot \vec{b} = (a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}) \cdot (a_2 \hat{i} + b_2 \hat{j} + c_2 \hat{k}) \)

\( = a_1 a_2 \hat{i} \cdot \hat{i} + a_1 b_2 \hat{i} \cdot \hat{j} + a_1 c_2 \hat{i} \cdot \hat{k} \) \( + b_1 a_2 \hat{j} \cdot \hat{i} + b_1 b_2 \hat{j} \cdot \hat{j} + b_1 c_2 \hat{j} \cdot \hat{k} \)

\( + c_1 a_2 \hat{k} \cdot \hat{i} + c_1 b_2 \hat{k} \cdot \hat{j} + c_1 c_2 \hat{k} \cdot \hat{k} \)


\( \vec{a} \cdot \vec{b} = a_1 a_2 + b_1 b_2 + c_1 c_2 \)

 

Cross Product: The cross product of two vectors can be calculated using the determinant of a 3 × 3 matrix formed with unit vectors i, j, and k in the first row, and the components of the two vectors in the second and third rows. For vectors \( \vec{a} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k} \) and b \( \vec{a} = a_2 \hat{i} + b_2 \hat{j} + c_2 \hat{k} \), the result is a new vector that is perpendicular to both vectors, and the direction is determined using the right-hand rule. 

The formula to calculate the cross-product is:

\( \vec{a} \times \vec{b} = \hat{i}(b_1 c_2 - b_2 c_1) - \hat{j}(a_1 c_2 - a_2 c_1) + \hat{k}(a_1 b_2 - a_2 b_1) \)
 

Here, \( \hat{i}, \; \hat{j}, \; \hat{k} \)  are the standard unit vectors along the x, y, and axis, respectively.

The following are the properties of the multiplication of vectors, which are divided based on the type of multiplication, which is the dot product and the cross product

1. The key properties of the dot product of two vectors \( \vec{a}, and \; \vec{b} \) is
 

• Commutativity: \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \)
   
• Distributivity over vector addition:\( \vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} \)
   
• Scalar multiplication: \( (k \vec{a}) \cdot \vec{b} = k (\vec{a} \cdot \vec{b}) = \vec{a} \cdot (k \vec{b}) \), here, k is a scalar.
   
• Dot product with itself: \( \vec{a} \cdot \vec{a} = |\vec{a}|^2 \)
   
• Relationship with angle: \( \vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \cos \theta \), where \(\theta\) is the angle between \( \vec{a} \) and \(\vec{b} \).
   
• Orthogonality: \( \vec{a} \cdot \vec{b} = 0 \) if and only if \(\vec{a}\) and \(\vec{b}\) are orthogonal (perpendicular), unless one of them is a zero vector.
   
• Projection: The scalar projection of vector a onto vector b is given by: \( \frac{\vec{a} \cdot \vec{b}}{\vec{|b|}} \), where \(b ≠ 0 \).

2. The key properties of the cross product of two vectors \(\vec{a} \) and \(\vec{b}\)
 

• Anti-commutativity: \(\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a}) \)
   
• Distributivity: \( \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} \)
   
• Cross product with itself: \( \vec{a} \cdot \vec{a} = 0 \), and in the magnitude formula, \(sin(0°) = 0\).
   
• Cross product with zero vector: \( \vec{a} \cdot \vec{0} = 0 \) and \( \vec{0} \cdot \vec{a} = 0 \).
   
• Magnitude: \( |\vec{a} \times \vec{b}| = |\vec{a}| \, |\vec{b}| \, \sin \theta \), here is the angle between the vectors.
   
• Parallel vectors: \( \vec{a} \cdot \vec{b} = 0 \)
   
• Unit vector:  The cross products of the standard unit vectors i, j, k follow a cyclic pattern based on the right-hand rule: \( \hat{i} \times \hat{j} = \hat{k}, \quad \hat{j} \times \hat{k} = \hat{i} \) and \( \hat{k} \times \hat{i} = \hat{j} \).

These properties are anti-commutative properties: \( \hat{i} \times \hat{j} = -\hat{k}, \quad \hat{j} \times \hat{k} = -\hat{i} \) and \( \hat{k} \times \hat{i} = -\hat{j} \). Also, the cross product of any unit vector with itself is zero: \( \hat{i} \times \hat{i} = 0, \quad \hat{j} \times \hat{j} = 0 \) and \( \hat{k} \times \hat{k} = 0 \) 

The given tips are useful, in making the multiplications of vectors an easier and efficient process.
 

• See whether to use dot product (scalar) or cross product (vector) based on the problem.
   
• Break vectors into components for easier calculations.
   
• Apply formulas carefully. Remember \( \vec{a} \cdot \vec{b} = |\vec{a}| \, |\vec{b}| \cos \theta \) and \( \vec{a} \times \vec{b} = |\vec{a}| \, |\vec{b}| \, \sin\theta \, \hat{n} \).
   
• For cross products, ensure the resulting vector follows the right-hand rule.
   
• Expand terms methodically step by step and verify each component to avoid mistakes.

Vector multiplication, through dot and cross products, has practical uses like:

1. Work Done: A dot product helps calculate work by finding the force component in the direction of

motion (e.g., pushing a box).

2. Torque: Cross product determines the rotational force (torque) crucial for tools and machinery (e.g., using a wrench).
 

3. Computer Graphics Lighting: The Dot product is used to calculate the angle between a light source and a surface. So it helps to determine how much light the surface receives, making it possible to create realistic lighting and shading on 3D graphics.
 

4. Magnetic Force: Cross product determines the force on moving charges in magnetic fields, essential for electromagnetism (e.g., motors).

5. Angular Momentum: The cross product is used to calculate angular momentum, which is key in understanding the rotational motion of objects, such as spinning wheels, planets, or satellites.