Physics 1D03 - Lecture 31 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 32 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 33 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 34 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 35 Vector Addition: Vector + Vector = Vector Triangle Method Parallelogram Method

Physics 1D03 - Lecture 36 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 newtons c)not enough information to tell

Physics 1D03 - Lecture 37 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)100N b)-100N c) 500N

Physics 1D03 - Lecture 38 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 ( x, y ) 0 x y

Physics 1D03 - Lecture 39 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 vyvy vxvx

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

Physics 1D03 - Lecture 311 x y z Is this a right-handed coordinate system? Does it matter?

Physics 1D03 - Lecture 312 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 313 A vector can be written in terms of its components: i j Ax iAx i Ay jAy j Ay jAy j Ax iAx i

Physics 1D03 - Lecture 314 Addition again: AxAx AyAy ByBy BxBx ByBy BxBx AyAy AxAx CxCx CyCy If A + B = C, then: Three scalar equations from one vector equation! Tail to Head

Physics 1D03 - Lecture 315 In components (2-D for simplicity) : The unit-vector notation leads to a simple rule for the components of a vector sum: Eg: A=2i+4j B=3i-5j A+B = 5i-j A - B = -i+9j

Physics 1D03 - Lecture 316 Magnitude : the “length” of a vector. Magnitude is a scalar. In terms of components: On the diagram, v x = v cos  v y = v sin  x y vyvy vxvx  e.g., Speed is the magnitude of velocity: velocity = v ; speed = |v| = v

Physics 1D03 - Lecture 317 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: