Chapter 3 Vectors Introduction Addition of Vectors Subtraction
Chapter 3 : Vectors - Introduction - Addition of Vectors - Subtraction of Vectors - Scalar Multiplication of Vectors - Components of Vectors - Magnitude of Vectors - Product of 2 Vectors - Application of Scalar/Dot Product & Cross Product
Introduction Vectors • Has magnitude (represent by length of arrow). • direction (direction of the arrow either to the right, left, etc). • Eg: move the brick 5 m to the right Scalars • Has magnitude only. • Eg: move the brick 5 m.
Introduction Vectors Representation • Use an arrow connecting an initial point A to terminal point B. • Denote • Written as • Magnitude of
Introduction Vectors Negative • Vector in opposite direction, , but has same magnitude as .
Introduction Equal Vectors • If we have 2 vectors, with same magnitude & direction .
Addition of Vectors 1. The Triangle Law • Any 2 vectors can be added by joining the initial point of terminal point of. • Eg: to the
Addition of Vectors 2. The Parallelogram Law • If 2 vector quantities are represented by 2 adjacent sides of a parallelogram, then the diagonal of parallelogram will be equal to the summation of these 2 vectors. • Eg: • The parallelogram law is affected by the triangle law.
Addition of Vectors The sum of a number of vectors
Subtraction of Vectors • Is a special case of addition. • Eg:
Scalar Multiplication • k ; vector multiply with scalar, k. • . Parallel Vectors
Scalar Multiplication
Components of Vectors – Unit Vectors
Vectors in 2 Dimensional (R ) 2
Vectors in 3 Dimensional (R ) 3
Exercise : Draw the vector
Components of Vectors
Magnitude of Vectors 1. For Any Vector Example: Exercise:
Magnitude of Vectors 2. From one point to another point of vector Example:
Magnitude of Vectors Solution:
Do Exercise 3. 3 in Textbook page 70.
Unit Vectors Example:
Do Exercise 3. 4 in Textbook page 70.
Direction Angles & Cosines
Direction Angles & Cosines Example: Solution (i): Direction cosines Direction angles 90. 77
Direction Angles & Cosines Solution (ii) Direction cosines Direction angles
Do Exercise 3. 5 in Textbook page 72.
Do Tutorial 3 in Textbook page 85 : • No. 2 (i) • No. 3 (i) • No. 4 • No. 5 (iii) • No. 6 (i)
Operations of Vectors by Components Example: Solution:
Do Exercise 3. 6 in Textbook page 72.
Product of 2 Vectors Dot Product / Scalar Product Example: Solution:
Do Exercise 3. 7 in Textbook page 73.
Find Angle Between 2 Vectors Example: Solution:
Do Exercise 3. 8 in Textbook page 74.
Product of 2 Product Cross Product / Vector Product Example:
Product of 2 Product Cross Product / Vector Product Solution:
Do Exercise 3. 9 in Textbook page 74.
Find Angle Between 2 Vectors
Applications of Vectors • Projections • The Area of Triangle & Parallelogram • The Volume of Parallelepiped & Tetrahedron • Equations of Planes 3 • Parametric Equations of Line in R • Distance from a Point to the Plane
i. Projections
Scalar projection of b onto a: Vector projection of b onto a:
Example : i. Given and vector projection of b onto a ii. Find Solutions: given that . Find the scalar projection
ii. The Area of Triangle and Parallelogram
Example : Solutions:
Solutions:
iii. The Volume of Parallelepiped and Tetrahedron A parallelepiped is a three-dimensional formed by six parallelogram. • Define three vectors • To represent the three edges that meet at one vertex. • The volume of the parallelepiped is equal to the magnitude of their scalar triple product
• Volume of Parallelepiped • Volume of Tetrahedron
Example : Solution:
iv. Equations of Planes
Example: Solutions:
Example : Solutions:
v. Parametric Equations of a Line in
Parametric equations of a line : Cartesian equations :
Example : Solutions:
vi. Distance from a Point to the Plane
Example:
Solutions:
ii.
- Slides: 58