Vector

A vector is a mathematical object that is represented by an arrow in space, with a magnitude (length) and a direction. In mathematics, vectors are often represented as column matrices with their magnitude and direction components. Vectors can be added, subtracted, and multiplied by scalars to produce new vectors. The magnitude of a vector can be calculated using the Euclidean norm, and the direction of a vector can be represented using a unit vector.

Product of Vectors

The product of two vectors can be of two types: Scalar product (Dot product) and Vector product (Cross product).

Scalar Product (Dot Product): The scalar product (or dot product) of two vectors is a scalar quantity that represents the cosine of the angle between the two vectors. The dot product of two vectors a and b is represented by a.b and can be calculated as:

a.b = |a| |b| cos(θ) = a1b1 + a2b2 + a3b3 + ... + anbn,

where |a| and |b| represent the magnitudes of vectors a and b, respectively, and θ is the angle between the two vectors. The dot product has the following properties:

a) Commutative: The dot product is commutative, i.e., a.b = b.a.

b) Distributive: The dot product is distributive over vector addition, i.e., (a + b).c = a.c + b.c.

c) For any scalar n: The dot product is linear in both vectors, i.e., n(a.b) = (na).b = a.(nb).

d) Positive definite: The dot product is positive definite, i.e., a.a > 0 for a ≠ 0 and a.a = 0 for a = 0.

The dot product has various applications in physics and engineering, such as calculating work, power, and projection of vectors.

Vector Product (Cross Product): The vector product (or cross product) of two vectors is a vector quantity that represents the sine of the angle between the two vectors and the direction perpendicular to both vectors. The cross product of two vectors a and b is represented by a × b and can be calculated as:

a × b = |a| |b| sin(θ) n = [(a2b3 - a3b2), (a3b1 - a1b3), (a1b2 - a2b1)],

where |a| and |b| represent the magnitudes of vectors a and b, respectively, θ is the angle between the two vectors, and n is a unit vector perpendicular to both a and b. The direction of the cross product is given by the right-hand rule. The cross product has the following properties:

a) Anticommutative: The cross product is anticommutative, i.e., a × b = -(b × a).

b) Distributive: The cross product is distributive over vector addition, i.e., (a + b) × c = a × c + b × c.

c) For any scalar n: The cross product is linear in both vectors, i.e., n(a × b) = (n a) × b = a × (n b).

d) Orthogonal: The cross product a × b is perpendicular to both a and b.

The cross product has various applications in physics and engineering, such as finding the area of a parallelogram, calculating torque and angular momentum, and finding the normal vector of a plane.

Geometrical Interpretation of Vector Product: The cross product of two vectors can be interpreted geometrically as the area of the parallelogram spanned by the two vectors. The magnitude of the cross product of two vectors represents the area of the parallelogram, and the direction of the cross product represents the normal vector to the plane containing the two vectors.

The geometrical interpretation of the cross product of two vectors is a crucial concept in vector calculus and is used in many fields including physics, engineering, and computer graphics. It provides a visual representation of the relationship between two vectors in three-dimensional space.

A cross product of two vectors a and b is a third vector, c, that is perpendicular to both a and b. This means that the direction of c is perpendicular to the plane containing the vectors a and b. The magnitude of the cross product, |c|, represents the area of the parallelogram spanned by the two vectors a and b. This area is defined as the product of the magnitude of a, |a|, and the magnitude of b, |b|, and the sine of the angle between a and b, sin(θ):

|c| = |a| |b| sin(θ)

The direction of c is given by the right-hand rule, which states that if the fingers of the right hand are curled in the direction of a and then b, the thumb points in the direction of c. The cross product of a and b is then given by:

c = a × b = |a| |b| sin(θ) n

where n is a unit vector perpendicular to the plane containing a and b.

This geometrical interpretation of the cross product provides a useful tool for understanding the properties of the cross product, such as its distributive and communicative properties. It also helps in the visualization of the relationship between two vectors, especially in the context of physics and engineering, where the cross product is used to calculate torque and angular momentum.

Vector Product of Two Vectors in Determinant Form: The cross product of two vectors can also be calculated using the determinant form. Let a = [a1, a2, a3] and b = [b1, b2, b3], then the cross product can be represented as:

| i j k |

|a1 a2 a3|

|b1 b2 b3|

Expression for sin(θ): The sine of the angle between two vectors can be calculated from the cross product as:

sin(θ) = |a × b| / (|a| |b|)

Unit Vectors: A unit vector is a vector of magnitude 1 that represents the direction of a vector. Unit vectors are often used to represent vectors in space, as any vector can be represented by a linear combination of unit vectors. The unit vectors in the x, y, and z directions can be represented as:

i = [1, 0, 0]

j = [0, 1, 0]

k = [0, 0, 1]

Vector Equation of a Straight Line: The vector equation of a straight line can be represented as a combination of a point on the line and a direction vector. Given a point P on the line and a direction vector d, the vector equation of the line can be represented as:

r = P + td, where t is a scalar parameter that can be used to obtain different points on the line.

Examples:

Example of dot product: Consider two vectors a = [1, 2, 3] and b = [2, 3, 4]. Then the dot product of the two vectors can be calculated as:

a.b = 1 * 2 + 2 * 3 + 3 * 4 = 20

Example of cross product: Consider two vectors a = [1, 2, 3] and b = [2, 3, 4]. Then the cross product of the two vectors can be calculated as:

a × b = [(2 * 4 - 3 * 3), (3 * 1 - 1 * 4), (1 * 3 - 2 * 2)] = [-1, -2, -1]

Example of vector equation of a straight line: Consider a line passing through the point (1, 2, 3) with direction vector [1, 1, 1]. Then the vector equation of the line can be written as:

r = [1, 2, 3] + t[1, 1, 1]

Example of unit vectors: The unit vectors in the x, y, and z directions can be represented as:

i = [1, 0, 0]

j = [0, 1, 0]

k = [0, 0, 1]

These unit vectors can be used to represent any vector in space by taking linear combinations of them.