Vectors - Analytic Description Of Vectors

Vectors - Analytic description of vectors

Definition of a vector
Components of a vector
Magnitude of a vector
Direction of a vector
Standard basis vectors

Definition of a vector

A vector is a quantity that has both magnitude and direction.
It can be represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction of the vector.
Vectors are commonly used to represent physical quantities such as force, velocity, and displacement.

Components of a vector

A vector can be represented in terms of its components.
The components of a vector represent the projections of the vector onto the coordinate axes.
For a vector v in three-dimensional space, its components are denoted as v = (v1, v2, v3), where v1, v2, and v3 are the projections of v onto the x, y, and z axes respectively.

Magnitude of a vector

The magnitude of a vector is the length or size of the vector.
It is denoted by ||v|| or |v|.
For a vector v = (v1, v2, v3), the magnitude is calculated using the formula: ||v|| = sqrt(v1^2 + v2^2 + v3^2).
The magnitude of a vector is always a non-negative value.

Direction of a vector

The direction of a vector can be represented in various ways.
It can be described using angles or by specifying its direction cosines or direction ratios.
Two vectors are said to be parallel if they have the same direction or opposite direction.
The direction of a vector can also be specified by its bearing or inclination.

Standard basis vectors

In three-dimensional space, there are three standard basis vectors: i, j, and k.
The vector i = (1, 0, 0) represents the unit vector along the x-axis.
The vector j = (0, 1, 0) represents the unit vector along the y-axis.
The vector k = (0, 0, 1) represents the unit vector along the z-axis.
Any vector in three-dimensional space can be expressed as a linear combination of these standard basis vectors.

Example: Finding the components of a vector

Consider a vector v = (-2, 3, 1).
The components of v are -2 along the x-axis, 3 along the y-axis, and 1 along the z-axis.
Thus, the components of v are v1 = -2, v2 = 3, and v3 = 1.

Example: Calculating the magnitude of a vector

Let’s find the magnitude of vector v = (3, -4, 2).
Using the formula for magnitude, ||v|| = sqrt(3^2 + (-4)^2 + 2^2) = sqrt(9 + 16 + 4) = sqrt(29).
Therefore, the magnitude of vector v is sqrt(29).

Example: Describing the direction of a vector

Consider a vector v = (1, -1, 2).
The direction of v can be described by its direction cosines, which are the cosines of the angles it makes with the positive x, y, and z axes.
The direction cosines of v are cos(alpha) = 1/sqrt(6), cos(beta) = -1/sqrt(6), and cos(gamma) = 2/sqrt(6).
Therefore, the direction of v is given by the angles alpha, beta, and gamma such that cos(alpha) = 1/sqrt(6), cos(beta) = -1/sqrt(6), and cos(gamma) = 2/sqrt(6).

Example: Expressing a vector in terms of standard basis vectors

Let’s express the vector v = (4, -5, 3) in terms of the standard basis vectors i, j, and k.
v = 4i - 5j + 3k.
Therefore, the vector v can be expressed as a linear combination of the standard basis vectors.

Operations on Vectors

Addition of vectors
Subtraction of vectors
Scalar multiplication of vectors
Dot product of vectors
Cross product of vectors

Addition of Vectors

When adding vectors, their corresponding components are added together.
The result is a vector with components equal to the sum of the corresponding components of the original vectors.
For example, if vector a = (a1, a2, a3) and vector b = (b1, b2, b3), then vector c = a + b = (a1 + b1, a2 + b2, a3 + b3).

Subtraction of Vectors

When subtracting vectors, their corresponding components are subtracted from each other.
The result is a vector with components equal to the difference of the corresponding components of the original vectors.
For example, if vector a = (a1, a2, a3) and vector b = (b1, b2, b3), then vector c = a - b = (a1 - b1, a2 - b2, a3 - b3).

Scalar Multiplication of Vectors

Scalar multiplication of a vector involves multiplying the vector by a scalar (a real number).
The result is a vector whose components are obtained by multiplying each component of the original vector by the scalar.
For example, if vector a = (a1, a2, a3) and scalar k, then vector c = ka = (ka1, ka2, ka3).

Dot Product of Vectors

The dot product of two vectors a and b is a scalar quantity.
It is calculated by multiplying the corresponding components of the vectors and then summing them.
The dot product is denoted by a · b or a ⋅ b.
The dot product is given by the formula: a · b = a1b1 + a2b2 + a3b3.

Example: Dot Product of Vectors

Let vector a = (1, 2, 3) and vector b = (-1, 2, -3).
The dot product is calculated as a · b = 1*(-1) + 22 + 3(-3) = -1 + 4 - 9 = -6.
Therefore, the dot product of a and b is -6.

Properties of Dot Product

Commutative property: a · b = b · a
Distributive property: a · (b + c) = a · b + a · c
Scalar multiplication: k(a · b) = (ka) · b = a · (kb)
Magnitude: |a · b| = ||a|| ||b|| cos(theta), where theta is the angle between a and b

Cross Product of Vectors

The cross product of two vectors a and b is a vector quantity.
It is calculated by taking the determinant of a matrix formed by the components of the vectors.
The cross product is denoted by a x b.
The cross product is given by the formula: a x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).

Example: Cross Product of Vectors

Let vector a = (1, 2, 3) and vector b = (-1, 2, -3).
The cross product is calculated as a x b = (2*(-3) - 32, 3(-1) - 1*(-3), 12 - 2(-1)) = (-12, 0, 4).
Therefore, the cross product of a and b is (-12, 0, 4).

Properties of Cross Product

Anti-commutative property: a x b = -(b x a)
Distributive property: a x (b + c) = a x b + a x c
Scalar multiplication: k(a x b) = (ka) x b = a x (kb)
Magnitude: ||a x b|| = ||a|| ||b|| sin(theta), where theta is the angle between a and b

Vector Equations

A vector equation is an equation involving vectors.
It represents a relationship between vectors in terms of their components.
Vector equations can be solved by considering the components separately.
For example, the vector equation a + b = c can be solved by equating the components: a1 + b1 = c1, a2 + b2 = c2, a3 + b3 = c3.

Magnitude and Direction of a Vector Equation

The magnitude of a vector equation can be found by taking the magnitude of both sides.
For example, if a + b = c, then |a + b| = |c|.
The direction of a vector equation can be determined by finding the direction of the resultant vector.
The direction of the resultant vector can be described using angles or direction cosines.

Linear Dependence and Independence of Vectors

A set of vectors is said to be linearly dependent if one or more of the vectors can be expressed as a linear combination of the other vectors.
A set of vectors is said to be linearly independent if none of the vectors can be expressed as a linear combination of the other vectors.
Linear dependence can be determined by setting up a system of equations and solving for the coefficients of the linear combination.
Linear independence can be determined by the existence of a non-trivial solution to the system of equations.

Orthogonal Vectors

Two vectors are said to be orthogonal if their dot product is zero.
Orthogonal vectors are also known as perpendicular vectors.
If vector a · b = 0, then a and b are orthogonal.
Orthogonal vectors can be used to find angles and distances in geometry and physics.

Parallel Vectors

Two vectors are said to be parallel if they have the same or opposite direction.
Parallel vectors can be scalar multiples of each other.
If vector a = kb, where k is a scalar, then a and b are parallel.
Parallel vectors have the same or opposite direction cosines.

Projection of a Vector

The projection of a vector a onto a vector b is a scalar quantity.
It represents the length of the shadow of a when it is cast onto the line determined by b.
The projection of a onto b is given by the formula: proj_b a = (a · b) / |b|.
The projection vector can be calculated using the formula: **proj_ba = (proj_ba / |b|) b.

Example: Projection of a Vector

Let vector a = (3, 4) and vector b = (1, 2).
The projection of a onto b is calculated as proj_b a = (a · b) / |b| = (31 + 42) / sqrt(1^2 + 2^2) = (11 / sqrt(5)).
Therefore, the projection of a onto b is (11 / sqrt(5)).

Cross Product and Area of Parallelogram

The cross product of two vectors can be used to find the area of a parallelogram.
The magnitude of the cross product of vectors a and b, denoted as a x b, is equal to the area of the parallelogram formed by a and b.
The direction of the cross product gives the sense of rotation of the parallelogram.
The area of the parallelogram can be calculated using the formula: A = ||a x b||.

Example: Cross Product and Area of Parallelogram

Let vector a = (2, 3, 4) and vector b = (1, -2, 3).
The cross product of a and b is calculated as a x b = (18, 5, -7).
The magnitude of the cross product is ||a x b|| = sqrt(18^2 + 5^2 + (-7)^2) = sqrt(454).
Therefore, the area of the parallelogram formed by a and b is sqrt(454).

Applications of Vectors

Vectors have numerous applications in various fields of science and engineering.
They are used to represent physical quantities such as force, velocity, and acceleration.
Vectors are essential in analyzing and understanding motion, both linear and rotational.
They are used in applications such as navigation, robotics, and computer graphics.
Vectors are also used in mathematical fields such as linear algebra and calculus.