Vectors

1. Vectors • Scalars and Vectors • Vector Components and Arithmetic • Vectors in 3 Dimensions • Unit vectors i, j, k Serway and Jewett Chapter 3 Physics 1D03 - Lecture 3

2. Physical quantities are classified as scalars, vectors, etc. Scalar : described by a real number with units examples: mass, charge, energy . . . Vector : described by a scalar (its magnitude) and a direction in space examples: displacement, velocity, force . . . Vectors have direction, and obey different rules of arithmetic. Physics 1D03 - Lecture 3

3. Notation • Scalars : ordinary or italic font (m, q, t . . .) • Vectors : - Boldface font (v, a, F . . .) - arrow notation - underline (v, a, F . . .) • Pay attention to notation : “constant v” and “constant v” mean different things! Physics 1D03 - Lecture 3

4. Magnitude : a scalar, is the “length” of a vector. e.g., Speed, v = |v| (a scalar), is the magnitude of velocity v Multiplication: scalar  vector = vector Later in the course, we will use two other types of multiplication: the “dot product” , and the “cross product”. Physics 1D03 - Lecture 3

5. Vector Addition: Vector + Vector = Vector Parallelogram Method Triangle Method Physics 1D03 - Lecture 3

6. Concept Quiz Two students are moving a refrigerator. One pushes with a force of 200 newtons, the other with a force of 300 newtons. Force is a vector. The total force they (together) exert on the refrigerator is: • equal to 500 newtons • equal to newtons • not enough information to tell Physics 1D03 - Lecture 3

7. Concept Quiz Two students are moving a refrigerator. One pushes with a force of 200 newtons (in the positive direction), the other with a force of 300 newtons in the opposite direction. What is the net force ? a)100Nb)-100Nc) 500N Physics 1D03 - Lecture 3

8. Coordinate Systems In 2-D : describe a location in a plane y • by polar coordinates : • distance r and angle  • by Cartesian coordinates : • distances x, y, parallel to axes with: x=rcosθ y=rsinθ ( x , y ) r y  x 0 x Physics 1D03 - Lecture 3

9. y vy vx x Components • define the axes first • are scalars • axes don’t have to be horizontal and vertical • the vector and its components form a right triangle with the vector on the hypotenuse Physics 1D03 - Lecture 3

10. z y y x x z 3-D Coordinates (location in space) We use a right-handed coordinate system with three axes: Physics 1D03 - Lecture 3

11. x y z Is this a right-handed coordinate system? Does it matter? Physics 1D03 - Lecture 3

12. A unit vector u or is a vector with magnitude 1 : (a pure number, no units) Define coordinate unit vectorsi, j, k along the x, y, z axis. z k j y i x Unit Vectors Physics 1D03 - Lecture 3

13. Ayj Ayj j i Axi Axi A vector can be written in terms of its components: Physics 1D03 - Lecture 3

14. By Ay Ax Bx By Cy Bx Ay Ax Cx Addition again: IfA + B = C , then: Tail to Head Three scalar equations from one vector equation! Physics 1D03 - Lecture 3

15. The unit-vector notation leads to a simple rule for the components of a vector sum: In components (2-D for simplicity) : Eg: A=2i+4j B=3i-5j A+B = 5i-j A - B = -i+9j Physics 1D03 - Lecture 3

16. y vy  vx x Magnitude : the “length” of a vector. Magnitude is a scalar. e.g., Speed is the magnitude of velocity: velocity = v ; speed = |v| = v In terms of components: On the diagram, vx = v cos vy = v sin Physics 1D03 - Lecture 3

17. Summary • vector quantities must be treated according to the rules of vector arithmetic • vectors add by the triangle rule or parallelogram rule(geometric method) • a vector can be represented in terms of its Cartesian components using the “unit vectors” i, j, kthese can be used to add vectors (algebraic method) • if and only if: Physics 1D03 - Lecture 3