Vectors - Problems - Parallelogram Geometry
Recap of vectors in 2-dimensional space
Introduction to parallelogram geometry
Definition of a parallelogram
Properties of a parallelogram
Opposite sides are parallel
Opposite angles are congruent
Diagonals bisect each other
Application of vectors in parallelogram geometry
Vector Addition and Subtraction
Review of vector addition and subtraction
Geometric interpretation of vector addition and subtraction
Calculation of resultant vectors
Finding magnitude and direction of resultant vectors
Example: Adding and subtracting vectors in 2-dimensional space
Scaling of Vectors
Definition of scalar multiplication
Scaling vectors by a scalar
Properties of scalar multiplication
Distributive property
Associative property
Calculation of scaled vectors
Example: Scaling vectors in 2-dimensional space
Dot Product of Vectors
Definition of dot product of vectors
Geometric interpretation of dot product
Calculation of dot product
Properties of dot product
Commutative property
Distributive property
Calculation of angle between vectors using dot product
Example: Finding dot product and angle between vectors
Cross Product of Vectors
Definition of cross product of vectors
Geometric interpretation of cross product
Calculation of cross product
Properties of cross product
Anticommutative property
Distributive property
Calculation of the magnitude and direction of the cross product
Example: Finding cross product and its properties
Vector Projection
Introduction to vector projection
Calculation of the projection of a vector onto another vector
Calculation of the component of a vector along another vector
Geometric interpretation of vector projection
Example: Finding vector projections in 2-dimensional space
Vector Equation of a Line
Definition of a vector equation of a line
Formulation of vector equation using a position vector and a direction vector
Calculation of the direction vector from two points on the line
Conversion between vector equation and parametric equations of a line
Example: Finding vector equation and parametric equation of a line
Scalar Triple Product
Definition of scalar triple product
Calculation of scalar triple product
Properties of scalar triple product
Antisymmetry
Distributive property
Geometric interpretation of scalar triple product
Example: Finding scalar triple product and its properties
Vector Triple Product
Definition of vector triple product
Calculation of vector triple product
Properties of vector triple product
Antisymmetry
Distributive property
Geometric interpretation of vector triple product
Example: Finding vector triple product and its properties
Projection of one Vector onto Plane
Introduction to projection of a vector onto a plane
Calculation of the projection of a vector onto a plane
Geometric interpretation of vector projection onto a plane
Example: Finding the projection of a vector onto a plane
Vectors in 3-Dimensional Space
Introduction to vectors in 3-dimensional space
Definition of a vector in 3D
Representation of a vector in 3D using coordinate form
Calculation of magnitude and direction of a vector in 3D
Example: Finding magnitude and direction of a vector in 3D
Vector Addition in 3D
Review of vector addition in 3D
Geometric interpretation of vector addition in 3D
Calculation of resultant vectors in 3D
Example: Adding vectors in 3D using coordinate form
Vector Dot Product in 3D
Extension of dot product to vectors in 3D
Calculation of dot product in 3D
Properties of dot product in 3D
Commutative property
Distributive property
Geometric interpretation of dot product in 3D
Example: Finding dot product of vectors in 3D
Vector Cross Product in 3D
Extension of cross product to vectors in 3D
Calculation of cross product in 3D
Properties of cross product in 3D
Anticommutative property
Distributive property
Geometric interpretation of cross product in 3D
Example: Finding cross product of vectors in 3D
Equations of Lines and Planes in 3D
Parametric equations of a line in 3D
Vector equation of a line in 3D
Finding the direction vector and position vector of a line
Equation of a plane in 3D
Example: Finding equations of lines and planes in 3D
Angle Between Two Vectors in 3D
Calculation of the angle between two vectors in 3D using dot product
Geometric interpretation of the angle between two vectors in 3D
Example: Finding the angle between two vectors in 3D
Projection of One Vector onto Another in 3D
Calculation of the projection of one vector onto another in 3D
Geometric interpretation of vector projection in 3D
Example: Finding the projection of one vector onto another in 3D
Application of Vectors in Physics
Introduction to the application of vectors in physics
Newton’s laws of motion and vectors
Vector forces and acceleration
Application of vectors in projectile motion
Example: Solving physics problems using vectors
Application of Vectors in Engineering
Introduction to the application of vectors in engineering
Vector forces and equilibrium
Vector forces and structural analysis
Application of vectors in electrical circuits
Example: Applying vectors in engineering problems
Review and Practice Problems
Recap of vectors and their properties
Review of vector operations
Practice problems involving vectors in 2D and 3D
Solution techniques for vector problems
Example: Solving practice problems on vectors
Properties of Parallelograms
Opposite sides are congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
The sum of the squares of the lengths of the four sides is equal to the sum of the squares of the lengths of the diagonals
Example: Applying Parallelogram Properties
Given a parallelogram ABCD with side lengths AB = 4 cm and BC = 6 cm.
Find the length of diagonals AC and BD.
Calculate the measure of angles BAC and CDA.
Prove that the sum of the squares of the lengths of the sides is equal to the sum of the squares of the lengths of the diagonals.
Definition of a Rhombus
Definition of a rhombus
Properties of a rhombus
All sides are congruent
Opposite angles are congruent
Diagonals are perpendicular bisectors
Diagonals bisect the angles
Geometric representation of a rhombus
Example: Solving Problems Involving Rhombi
Given a rhombus ABCD with side length 8 cm and one diagonal of length 12 cm.
Calculate the length of the other diagonal.
Find the measure of each angle of the rhombus.
Prove that the diagonals are perpendicular bisectors.
Definition of a Rectangle
Definition of a rectangle
Properties of a rectangle
All angles are right angles
Opposite sides are congruent
Diagonals are congruent and bisect each other
Geometric representation of a rectangle
Example: Working with Rectangles
Given a rectangle ABCD with length 10 cm and width 6 cm.
Calculate the diagonals of the rectangle.
Find the measure of each angle of the rectangle.
Show that the diagonals bisect each other.
Definition of a Square
Definition of a square
Properties of a square
All sides are congruent
All angles are right angles
Diagonals are congruent and bisect each other at right angles
Geometric representation of a square
Example: Solving Problems Involving Squares
Given a square ABCD with side length 5 cm.
Calculate the length of the diagonals.
Find the measure of each angle of the square.
Prove that the diagonals are congruent and bisect each other at right angles.
Introduction to Trapezoids
Definition of a trapezoid
Properties of a trapezoid
One pair of opposite sides is parallel
Non-parallel sides are not congruent
Diagonals do not bisect each other
Geometric representation of a trapezoid
Example: Analyzing Trapezoids
Given a trapezoid ABCD with parallel sides AB = 6 cm and CD = 8 cm, with height h = 4 cm.
Calculate the measures of angles A and B.
Find the lengths of the diagonals AC and BD.
Prove that the diagonals do not bisect each other.