Vectors - Important identities and inequalities

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  • Introduction to vectors
  • Types of vectors: position, displacement, velocity
  • Vector addition and subtraction
  • Scalar multiplication of vectors
  • Unit vectors and direction angles
  • Magnitude and direction of a vector
  • Dot product of vectors
  • Properties of dot product
  • Applications of dot product: finding angle between vectors, projection of a vector
  • Cross product of vectors
  • Properties of cross product
  • Applications of cross product: finding area of parallelogram, finding direction of vector perpendicular to two vectors
  • Vector projection
  • Vector component
  • Vector equation of a line
  • Parametric equations of a line
  • Distance between a point and a line
  • Vector equation of a plane
  • Scalar equation of a plane
  • Distance between a point and a plane
  • Angle between two planes
  • 3D vectors
  • Scalar triple product
  • Properties of scalar triple product
  • Applications of scalar triple product: finding volume of parallelepiped
  • Vector triple product
  • Properties of vector triple product
  • Applications of vector triple product: finding volume of tetrahedron, finding equation of a sphere
  • Vector identities
  • Addition and subtraction of vectors
  • Multiplication of a vector by a scalar
  • Distributive properties of vectors
  • Triangle inequality for vectors
  • Cauchy-Schwarz inequality
  • Triangle inequality for dot product
  • Triangle inequality for cross product
  • Orthogonal vectors
  • Normal vectors
  • Orthogonal projection
  • Orthogonal complement
  • Orthogonal basis
  1. Vector equation of a plane:
  • A plane in 3D can be represented by a vector equation.
  • The vector equation of a plane passing through a point P(x₁, y₁, z₁) with normal vector n = ai + bj + ck is given by:
    (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
  1. Scalar equation of a plane:
  • The scalar equation of a plane passing through a point P(x₁, y₁, z₁) with normal vector n = ai + bj + ck is given by: ax + by + cz = d, where d = ax₁ + by₁ + cz₁
  1. Distance between a point and a plane:
  • The distance between a point P(x₁, y₁, z₁) and a plane given by the equation ax + by + cz = d is given by: d = |ax₁ + by₁ + cz₁ - d| / √(a² + b² + c²)
  1. Angle between two planes:
  • The angle θ between two planes with normal vectors n₁ = a₁i + b₁j + c₁k and n₂ = a₂i + b₂j + c₂k is given by: cosθ = (a₁a₂ + b₁b₂ + c₁c₂) / √((a₁² + b₁² + c₁²)(a₂² + b₂² + c₂²))
  1. 3D vectors:
  • In 3D, a vector can be represented by the coordinates (x, y, z).
  • The magnitude of a 3D vector is given by: |v| = √(x² + y² + z²)
  • The direction of a 3D vector can be determined by its direction angles: α, β, γ, where cosα = x/|v|, cosβ = y/|v|, cosγ = z/|v|
  1. Scalar triple product:
  • The scalar triple product of three vectors a, b, and c is given by: [a, b, c] = a · (b x c)
  • It represents the volume of the parallelepiped formed by the three vectors.
  • The scalar triple product can be calculated using the determinant of a 3x3 matrix: [a, b, c] = | a₁ a₂ a₃ | | b₁ b₂ b₃ | | c₁ c₂ c₃ |
  1. Properties of scalar triple product:
  • [a, b, c] = -[b, a, c] = -[c, b, a]
  • [a, b, c] = -[b, c, a] = -[c, a, b]
  • [a, b, c] = 0 if a, b, and c are linearly dependent
  • [a, b, c] = |a| |b| |c| sinθ, where θ is the angle between b and c
  1. Applications of scalar triple product:
  • Finding the volume of a parallelepiped formed by three vectors.
  • Determining if three vectors are coplanar (if [a, b, c] = 0).
  • Finding the angle between two vectors using the dot product and scalar triple product.
  1. Vector triple product:
  • The vector triple product of three vectors a, b, and c is given by: [a, b, c] = a x (b x c)
  • It represents the vector perpendicular to the plane formed by the three vectors.
  • The vector triple product can be calculated using the cross product: [a, b, c] = (a · c) b - (a · b) c
  1. Properties of vector triple product:
  • [a, b, c] = -[b, a, c] = -[c, b, a]
  • [a, b, c] = -[b, c, a] = -[c, a, b]
  • [a, b, c] = 0 if a, b, and c are coplanar (if a, b x c, and c x a are collinear)
  1. Vector identities:
  • In addition to the basic operations of vector addition, subtraction, and scalar multiplication, there are several important vector identities.
  1. Addition and subtraction of vectors:
  • The addition of vectors follows the commutative property: a + b = b + a
  • The subtraction of vectors is the same as adding the negation: a - b = a + (-b)
  1. Multiplication of a vector by a scalar:
  • When multiplying a vector by a scalar, each component of the vector is multiplied by the scalar: k(a₁, a₂, a₃) = (ka₁, ka₂, ka₃)
  1. Distributive properties of vectors:
  • Distributive property of vector addition over scalar multiplication: k(a + b) = ka + kb
  • Distributive property of scalar multiplication over vector addition: (k + l)a = ka + la
  1. Triangle inequality for vectors:
  • The triangle inequality states that for any two vectors a and b, the magnitude of their sum is less than or equal to the sum of their magnitudes: |a + b| ≤ |a| + |b|
  1. Cauchy-Schwarz inequality:
  • The Cauchy-Schwarz inequality states that for any two vectors a and b, the dot product of the vectors is less than or equal to the product of their magnitudes: |a · b| ≤ |a| |b|
  1. Triangle inequality for dot product:
  • The triangle inequality for dot product states that for any two vectors a and b, the absolute value of their dot product is less than or equal to the product of their magnitudes: |a · b| ≤ |a| |b|
  1. Triangle inequality for cross product:
  • The triangle inequality for cross product states that for any two vectors a and b, the magnitude of their cross product is less than or equal to the product of their magnitudes: |a x b| ≤ |a| |b|
  1. Orthogonal vectors:
  • Two vectors a and b are orthogonal if and only if their dot product is zero: a · b = 0
  • Orthogonal vectors are also known as perpendicular vectors.
  1. Normal vectors:
  • A normal vector to a plane is a vector that is orthogonal to every vector lying within the plane.
  • The direction of a normal vector determines the orientation of the plane.
  • The equation of a plane can be determined using a normal vector and a point on the plane.