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 1 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 2 Notation Scalars : ordinary or italic font (m, q, t . . .) Vectors : - Boldface font (v, a, F . . .) - arrow notation ( v, a, F . . .) - underline (v, a, F . . .)
Pay attention to notation : constant v and constant v mean different things! Physics 1D03 - Lecture 3 3 Magnitude : a scalar, is the length of a vector. e.g., Speed, v = |v| (a scalar), is the magnitude of velocity v Multiplication: A scalar vector = vector 3 A 2
12 A Later in the course, we will use two other types of multiplication: the dot product , and the cross product. Physics 1D03 - Lecture 3 4 Vector Addition: Vector + Vector = Vector A e.g. A B C Triangle Method A
B C A B B Parallelogram Method A C A B B Physics 1D03 - Lecture 3 5 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: a) equal to 500 newtons b) equal to 200 2 300 2 newtons c) not enough information to tell Physics 1D03 - Lecture 3 6 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 7 Coordinate Systems In 2-D : describe a location in a plane y by polar coordinates : (x,y) distance r and angle r y by Cartesian coordinates : 0 x
x distances x, y, parallel to axes with: x=rcos y=rsin Physics 1D03 - Lecture 3 8 Components define the axes first y v x , v y , (and v z ) are scalars axes dont have to be horizontal and vertical v vy vx
x the vector and its components form a right triangle with the vector on the hypotenuse Physics 1D03 - Lecture 3 9 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: y | v | v x2 v y2 On the diagram, vx = v cos
v vy vx x vy = v sin Physics 1D03 - Lecture 3 10 3-D Coordinates (location in space) We use a right-handed coordinate system with three axes: z y y x
x z Physics 1D03 - Lecture 3 11 x Is this a right-handed coordinate system? Does it matter? y z Physics 1D03 - Lecture 3 12 Unit Vectors
A unit vector u or u is a vector with magnitude 1 : u 1 z (a pure number, no units) k Define coordinate unit vectors i, j, k along the x, y, z axis. j y i x Physics 1D03 - Lecture 3 13
A vector A can be written in terms of its components: A Ax i Ay j Az k A Ay j A Ay j j i Ax i Ax i
Physics 1D03 - Lecture 3 14 Addition again: If A + B = C , A Ay Ax B By Bx then: Cx Ax Bx Cy Ay By
Tail to Head Cz Az Bz Three scalar equations from one vector equation! B C A Ax By Cy Bx Ay
Cx Physics 1D03 - Lecture 3 15 The unit-vector notation leads to a simple rule for the components of a vector sum: A B C In components (2-D for simplicity) : A B ( Ax Bx ) i ( Ay By ) j Cx i Cy j Eg: A=2i+4j B=3i-5j A+B = 5i-j A - B = -i+9j Physics 1D03 - Lecture 3 16 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 A 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) C A B if and only if: Cx Ax Bx Cy Ay By Cz Az Bz Physics 1D03 - Lecture 3 17