Serway and Jewett Chapter 3 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

Vectors have direction, and obey different rules of arithmetic. 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

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

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

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

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

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) 100 N B) -100 N C) 500 N Physics 1D03 - Lecture 3

Coordinate Systems In 2-D : describe a location in a plane 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 x Physics 1D03 - Lecture 3

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 x y vy vx Physics 1D03 - Lecture 3

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  x y vy vx  Physics 1D03 - Lecture 3

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

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

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

A vector can be written in terms of its components: Ay j Ay j j Ax i i Ax i Physics 1D03 - Lecture 3

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

In components (2-D for simplicity) : 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

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, k these can be used to add vectors (algebraic method) if and only if: Physics 1D03 - Lecture 3