Vector Algebra: Basic Operations

• Vector algebra deals with vectors and the rules to manipulate them
• Vectors are objects that have both magnitude and direction

Basic Vector Operations

• Vector addition and subtraction are fundamental operations in vector algebra
• Vector addition is commutative: a + b = b + a
• Vector addition is associative: (a + b) + c = a + (b + c)
• Vector subtraction can be defined as addition of the negative of a vector: a - b = a + (-b)
• Scalar multiplication involves multiplying a vector by a scalar (a real number)
• If c is a scalar and a a vector, then ca is a vector with magnitude |c| times the magnitude of a and with the same direction as a if c > 0, and opposite if c < 0
• Scalar multiplication is distributive: c(a + b) = ca + cb
• Scalar multiplication is associative: (c d)a = c(da) where c and d are scalars

Vector Components, Magnitude, and Unit Vectors

• In a Cartesian coordinate system, a vector can be expressed in terms of its components along the coordinate axes
• In 2D, a vector a can be written as a = a_x i + a_y j, where a_x and a_y are the x and y components, and i and j are the unit vectors along the x and y axes
• In 3D, a vector a can be written as a = a_x i + a_y j + a_z k, where a_x, a_y, and a_z are the x, y, and z components, and i, j, and k are the unit vectors along the x, y, and z axes
• The magnitude (or length) of a vector a is denoted by |a| or a
• In 2D, the magnitude is given by |a| = sqrt(a_x^2 + a_y^2)
• In 3D, the magnitude is given by |a| = sqrt(a_x^2 + a_y^2 + a_z^2)
• A unit vector is a vector with magnitude 1
• A unit vector in the direction of a is denoted by â and is calculated as â = a / |a|

Dot Product

• The dot product (also called scalar product) of two vectors a and b is a scalar quantity
• It is defined as a ⋅ b = |a| |b| cos(θ), where θ is the angle between a and b
• In component form, a ⋅ b = a_x * b_x + a_y * b_y (in 2D)
• In component form, a ⋅ b = a_x * b_x + a_y * b_y + a_z * b_z (in 3D)
• The dot product is commutative: a ⋅ b = b ⋅ a
• The dot product is distributive: a ⋅ (b + c) = a ⋅ b + a ⋅ c
• If a ⋅ b = 0 and neither a nor b is the zero vector, then a and b are orthogonal (perpendicular)
• The dot product can be used to find the angle between two vectors: cos(θ) = (a ⋅ b) / (|a| |b|)

Cross Product

• The cross product (also called vector product) of two vectors a and b is a vector quantity
• It is defined as a × b = |a| |b| sin(θ) n, where θ is the angle between a and b, and n is a unit vector perpendicular to both a and b
• The direction of n is given by the right-hand rule
• In component form, a × b = (a_y * b_z - a_z * b_y) i + (a_z * b_x - a_x * b_z) j + (a_x * b_y - a_y * b_x) k (in 3D)
• The cross product is anti-commutative: a × b = - (b × a)
• The cross product is distributive: a × (b + c) = a × b + a × c
• If a × b = 0 and neither a nor b is the zero vector, then a and b are parallel
• The magnitude of a × b is equal to the area of the parallelogram formed by a and b

Applications

• Vectors are used extensively in physics for representing forces, velocities, and accelerations
• They are also used in computer graphics for representing positions, orientations, and transformations of objects
• Vector algebra is used in engineering for analyzing structures, circuits, and control systems
• Navigation: Vectors are crucial for representing directions and magnitudes of movement, helping in calculating routes and distances
• Game development: Vectors are used to represent position, velocity, and acceleration of game entities. They are essential for physics simulations, character movement, and creating interactive environments.

Explore fundamental vector operations including addition, subtraction, and scalar multiplication. Understand commutative, associative, and distributive properties. Learn how scalars affect vector magnitude and direction.