Vectors - Problems - Angle between vectors

• Recap of vectors in 2D and 3D
• Definition of angle between two vectors
• Calculation of angle using dot product
• Example: Finding the angle between two vectors using dot product
• Properties of the angle between vectors

Vectors - Problems - Projection of vectors

• Definition of projection of a vector onto another vector
• Calculation of projection using dot product
• Example: Finding the projection of a vector onto another vector
• Properties of projection
• Applications of projection in physics and engineering

Vectors - Problems - Cross product

• Definition of cross product in 3D
• Calculation of cross product using determinant method
• Example: Finding the cross product of two vectors
• Properties of cross product
• Applications of cross product in physics and geometry

Vectors - Problems - Scalar triple product

• Definition of scalar triple product
• Calculation of scalar triple product using determinant method
• Example: Finding the scalar triple product of three vectors
• Properties of scalar triple product
• Applications of scalar triple product in geometry and physics

Vectors - Problems - Vector triple product

• Definition of vector triple product
• Calculation of vector triple product using cross product
• Example: Finding the vector triple product of three vectors
• Properties of vector triple product
• Applications of vector triple product in physics and mechanics

Matrices - Types of matrices

• Introduction to matrices
• Definition of diagonal, triangular, scalar, identity, and zero matrices
• Examples of each type of matrix
• Properties and characteristics of different types of matrices
• Applications of matrices in various fields

Matrices - Operations on matrices

• Addition and subtraction of matrices
• Scalar multiplication of matrices
• Matrix multiplication
• Properties and rules of matrix operations
• Examples illustrating matrix operations

Matrices - Determinants

• Definition of determinant of a matrix
• Calculation of determinants of 2x2 and 3x3 matrices
• Properties and rules of determinants
• Cramer’s rule for solving linear equations using determinants
• Applications of determinants in physics and computer science

Matrices - Inverse of a matrix

• Definition of inverse of a matrix
• Calculation of inverse of 2x2 and 3x3 matrices
• Properties and rules of matrix inverses
• Existence and uniqueness of matrix inverses
• Applications of matrix inverses in solving linear equations

Matrices - System of linear equations

• Introduction to systems of linear equations
• Representation of systems of linear equations using matrices
• Methods for solving systems of linear equations: Gaussian elimination and matrix inverses
• Applications of systems of linear equations in physics and engineering
• Example: Solving a system of linear equations using matrices

Vectors - Problems - Angle between vectors

• Recap of vectors in 2D and 3D
• Definition of angle between two vectors
• Calculation of angle using dot product
• Example: Finding the angle between two vectors using dot product
  • Let vector A = <a1, a2, a3> and vector B = <b1, b2, b3>
  • The dot product of A and B is given by A · B = a1b1 + a2b2 + a3*b3
  • The magnitudes of A and B are given by |A| = sqrt(a1^2 + a2^2 + a3^2) and |B| = sqrt(b1^2 + b2^2 + b3^2)
  • The angle between A and B is given by cos(theta) = (A · B) / (|A| * |B|)
• Properties of the angle between vectors
  • The angle between two vectors is always between 0 and 180 degrees
  • If the dot product of two vectors is positive, the angle between them is acute (less than 90 degrees)
  • If the dot product of two vectors is negative, the angle between them is obtuse (greater than 90 degrees)
  • If the dot product of two vectors is zero, the angle between them is 90 degrees (perpendicular)

Vectors - Problems - Projection of vectors

• Definition of projection of a vector onto another vector
• Calculation of projection using dot product
• Example: Finding the projection of a vector onto another vector
  • Let vector A = <a1, a2> and vector B = <b1, b2>
  • The projection of A onto B is given by proj_B A = ((A · B) / |B|^2) * B
  • The magnitude of the projection of A onto B is given by |proj_B A| = |A| * cos(theta), where theta is the angle between A and B
• Properties of projection
  • The projection of a vector onto itself is the vector itself
  • The projection of a vector onto a zero vector is the zero vector
• Applications of projection in physics and engineering
  • Finding the component of a force acting in a specific direction
  • Analyzing the motion of objects on inclined planes

Vectors - Problems - Cross product

• Definition of cross product in 3D
• Calculation of cross product using determinant method
• Example: Finding the cross product of two vectors
  • Let vector A = <a1, a2, a3> and vector B = <b1, b2, b3>
  • The cross product of A and B is given by A x B = <a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1>
• Properties of cross product
  • The cross product of two vectors is perpendicular to both vectors
  • The magnitude of the cross product represents the area of the parallelogram formed by the two vectors
• Applications of cross product in physics and geometry
  • Calculating torque in rotational mechanics
  • Determining the direction of magnetic fields in electromagnetism

Vectors - Problems - Scalar triple product

• Definition of scalar triple product
• Calculation of scalar triple product using determinant method
• Example: Finding the scalar triple product of three vectors
  • Let vector A = <a1, a2, a3>, vector B = <b1, b2, b3>, and vector C = <c1, c2, c3>
  • The scalar triple product of A, B, and C is given by A · (B x C) = det([[a1, a2, a3], [b1, b2, b3], [c1, c2, c3]])
• Properties of scalar triple product
  • The scalar triple product is equal to the volume of the parallelepiped formed by the three vectors
  • The scalar triple product can be used to determine whether three vectors are coplanar or not
• Applications of scalar triple product in geometry and physics
  • Computing volumes and areas in three-dimensional space
  • Determining the equilibrium of forces in statics problems

Vectors - Problems - Vector triple product

• Definition of vector triple product
• Calculation of vector triple product using cross product
• Example: Finding the vector triple product of three vectors
  • Let vector A = <a1, a2, a3>, vector B = <b1, b2, b3>, and vector C = <c1, c2, c3>
  • The vector triple product of A, B, and C is given by A x (B x C) = B * (A · C) - C * (A · B)
• Properties of vector triple product
  • The vector triple product is associative: A x (B x C) = (A x B) x C
  • The vector triple product is not commutative: A x (B x C) ≠ B x (A x C)
• Applications of vector triple product in physics and mechanics
  • Calculating the angular momentum of rotating objects
  • Solving problems involving torque and rotational motion

Vectors - Problems - Angle between vectors

• Recap of vectors in 2D and 3D
• Definition of angle between two vectors
• Calculation of angle using dot product
• Example: Finding the angle between two vectors using dot product
  • Let vector A = <a1, a2, a3> and vector B = <b1, b2, b3>
  • The dot product of A and B is given by A · B = a1b1 + a2b2 + a3*b3
  • The magnitudes of A and B are given by |A| = sqrt(a1^2 + a2^2 + a3^2) and |B| = sqrt(b1^2 + b2^2 + b3^2)
  • The angle between A and B is given by cos(theta) = (A · B) / (|A| * |B|)
• Properties of the angle between vectors
  • The angle between two vectors is always between 0 and 180 degrees
  • If the dot product of two vectors is positive, the angle between them is acute (less than 90 degrees)
  • If the dot product of two vectors is negative, the angle between them is obtuse (greater than 90 degrees)
  • If the dot product of two vectors is zero, the angle between them is 90 degrees (perpendicular)

Vectors - Problems - Projection of vectors

• Definition of projection of a vector onto another vector
• Calculation of projection using dot product
• Example: Finding the projection of a vector onto another vector
  • Let vector A = <a1, a2> and vector B = <b1, b2>
  • The projection of A onto B is given by proj_B A = ((A · B) / |B|^2) * B
  • The magnitude of the projection of A onto B is given by |proj_B A| = |A| * cos(theta), where theta is the angle between A and B
• Properties of projection
  • The projection of a vector onto itself is the vector itself
  • The projection of a vector onto a zero vector is the zero vector
• Applications of projection in physics and engineering
  • Finding the component of a force acting in a specific direction
  • Analyzing the motion of objects on inclined planes

Vectors - Problems - Cross product

• Definition of cross product in 3D
• Calculation of cross product using determinant method
• Example: Finding the cross product of two vectors
  • Let vector A = <a1, a2, a3> and vector B = <b1, b2, b3>
  • The cross product of A and B is given by A x B = <a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1>
• Properties of cross product
  • The cross product of two vectors is perpendicular to both vectors
  • The magnitude of the cross product represents the area of the parallelogram formed by the two vectors
• Applications of cross product in physics and geometry
  • Calculating torque in rotational mechanics
  • Determining the direction of magnetic fields in electromagnetism

Vectors - Problems - Scalar triple product

• Definition of scalar triple product
• Calculation of scalar triple product using determinant method
• Example: Finding the scalar triple product of three vectors
  • Let vector A = <a1, a2, a3>, vector B = <b1, b2, b3>, and vector C = <c1, c2, c3>
  • The scalar triple product of A, B, and C is given by A · (B x C) = det([[a1, a2, a3], [b1, b2, b3], [c1, c2, c3]])
• Properties of scalar triple product
  • The scalar triple product is equal to the volume of the parallelepiped formed by the three vectors
  • The scalar triple product can be used to determine whether three vectors are coplanar or not
• Applications of scalar triple product in geometry and physics
  • Computing volumes and areas in three-dimensional space
  • Determining the equilibrium of forces in statics problems

Vectors - Problems - Vector triple product

• Definition of vector triple product
• Calculation of vector triple product using cross product
• Example: Finding the vector triple product of three vectors
  • Let vector A = <a1, a2, a3>, vector B = <b1, b2, b3>, and vector C = <c1, c2, c3>
  • The vector triple product of A, B, and C is given by A x (B x C) = B * (A · C) - C * (A · B)
• Properties of vector triple product
  • The vector triple product is associative: A x (B x C) = (A x B) x C
  • The vector triple product is not commutative: A x (B x C) ≠ B x (A x C)
• Applications of vector triple product in physics and mechanics
  • Calculating the angular momentum of rotating objects
  • Solving problems involving torque and rotational motion