Chapter 3 Vectors Prof. Raymond Lee, revised 9-2-2010 1
Coordinate systems Used to describe a point s position in space Coordinate system consists of fixed reference point called origin specific axes with scales & labels instructions on how to label a point relative to origin & axes 2
Cartesian coordinate system Also called rectangular coordinate system x- & y- axes intersect at origin Points are labeled (x,y) (compare Fig. 3-9, p. 41) 3
Polar coordinate system Origin & reference line are noted Point is distance r from origin in direction of angle! that s CCW from reference line Points are labeled (r,!) (compare Fig. 3-8, p. 41) 4
Polar to Cartesian coordinates Based on forming right triangle from r &! x = r cos! y = r sin! (compare figure on p. 43) 5
Cartesian to polar coordinates r is hypotenuse &! an angle! must be CCW from +x axis for these equations to be valid 6
Cartesian example Cartesian coordinates of a point in xy plane are (x,y) = (-3.50, -2.50) m, as shown in figure. Find this point s polar coordinates. Solution: From Eq. 3-6, (SJ 2008, p. 54) 7
Vectors & scalars A scalar quantity is completely specified by a single value with an appropriate unit & has no direction. A vector quantity is completely described by a number & appropriate units + a direction. 8
Vector notation When handwritten, use a tilde underscore: A ~ When printed, vector will be in boldface: A When dealing with just the vector magnitude in print, use italics: A or A Vector magnitude has physical units & is always a + number 9
Vector example Particle travels from A to B along path shown by dotted red line, the scalar distance traveled Displacement is solid line from A to B, which is independent of path taken between the 2 points & is a vector (compare Fig. 3-2, p. 39) 10
Equality of 2 vectors 2 vectors are equal if they have same magnitude & direction A = B if A = B & they point along parallel lines All the vectors shown here are equal (compare Fig. 3-1, p. 38) 11
Adding vectors When adding vectors, must take into account their directions; units must be the same Graphical Methods Use scale drawings Algebraic Methods More convenient 12
Adding vectors graphically Choose a scale Draw 1st vector A with appropriate length & in direction specified w.r.t. a coordinate system Draw 2nd vector with appropriate length & in direction specified w.r.t. a coordinate system whose origin is end of vector A & is to coordinate system used for A 13
Adding vectors graphically, 2 Continue drawing the vectors tip-to-tail Resultant is drawn from tail of A to tip of last vector Measure length of R & its angle Use scale factor to convert R s length to actual magnitude (compare Fig. 3-3, p. 39) 14
Adding vectors graphically, 3 If have many vectors, repeat process until all are included Resultant is still drawn from origin of 1st vector to end of last vector (compare Fig. 3-4, p. 39) 15
Adding vectors, rules When we add 2 vectors, sum is independent of order of addition. This is commutative law of addition A + B = B + A (compare Fig. 3-8, p. 39) 16
Adding vectors, rules 2 When adding! 3 vectors, sum is independent of the way in which individual vectors are grouped This is called associative property of addition (A + B) + C = A + (B + C) (Fig. 3-3, p. 39) 17
Adding vectors, rules 3 When adding vectors, all vectors must have same units All vectors must measure same physical quantity (e.g., can t add a displacement to a velocity) 18
Negative of a vector Negative of a vector is defined as vector that, when added to original vector,! resultant of zero Represented as A A + (-A) = 0 vector has same magnitude, but points in opposite direction 19
Subtracting vectors Special case of vector addition If A B, then use A+(-B) Continue with standard vector addition procedure (compare Fig. 3-6, p. 40) 20
Multiplying or dividing vector by a scalar Result of multiplication or division is a vector Vector s magnitude is multiplied or divided by scalar If scalar > 0, direction of result is same as of original vector If scalar < 0, direction of result is opposite that of the original vector 21
Vector components A component is a part whole vector & is most useful with rectangular components Shown are projections of vector along the x- & y-axes (compare Fig. 3-8, p. 41) 22
Vector components A x & A y are component vectors of A They are vectors & follow all rules for vectors A x & A y are scalars & are called the components of A 23
Vector components, 2 Vector s x-component is its projection along x-axis y-component is vector s projection along y-axis Then which gives 24
Vector components, 3 y-component is moved to end of x-component Valid since any vector can be moved to itself without being affected This movement completes triangle (compare Fig. 3-8, p. 41) 25
Vector components, 4 Previous equations are valid only if! is measured with respect to the x-axis Components are legs of right triangle whose hypotenuse is A May still have to find " w.r.t. + x-axis 26
Vector components, 5 Components can be + or & will have same units as original vector Components signs depend on angle! (Fig. 3.13, p. 60) 27
Unit vectors Unit vector is dimensionless vector with magnitude = 1. Use unit vectors to specify a direction & have no other physical significance 28
Unit vectors, 2 Symbols î, ĵ, and kˆ represent unit vectors They form a set of mutually perpendicular (#) vectors (compare Fig. 3-13, p. 44) 29
Unit vector notation A x is same as A x, & A y is the same as A y, etc. Write complete vector as: (compare Fig. 3-14, p. 44) 30
Adding vectors using unit vectors Using R = A + B Then & so R x = A x + B x & R y = A y + B y 31
Trig function warning Component equations {A x = A cos(!) & A y = A sin(!)} apply only when angle is measured w.r.t. x-axis (preferably CCW from + x-axis). Resultant angle {tan(") = A y /A x } gives angle w.r.t. x-axis. Think of triangle being formed & corresponding!; then use appropriate trig functions 32
Adding vectors with unit vectors (SJ 2008 Fig. 3.16, p. 61) 33
Adding vectors using 3D unit vectors Using R = A + B R x = A x + B x, R y = A y + B y & R z = A z + B z etc. 34
Example: Taking a hike {SJ 2008, p. 63} A hiker starts a trip by first walking 25.0 km SE from her car. She stops & sets up her tent for the night. On 2nd day, she walks 40.0 km in a direction 60.0 N of E, at which point she discovers a forest ranger s tower. 35
hiking example, p. 2 (A) Determine components of hiker s displacement for each day. Solution: We conceptualize problem by drawing a sketch as in figure above. If we denote displacement vectors on 1st & 2nd days by A & B respectively, & use car as coordinate origin, we get vectors shown in figure. Drawing resultant R, we can now categorize this problem as an addition of 2 vectors. 36
hiking example, p. 3 Analyze this problem using vector components. Displacement A has magnitude = 25.0 km & is directed 45.0 below +x axis. From Eq. 3-5 (p. 41), its components are: value of A y indicates that hiker walks in y direction on 1st day. Signs of A x & A y also are evident from figure above. 37
hiking example, p. 4 2nd displacement B has a magnitude = 40.0 km & is 60.0 N of E. Its components are: 38
hiking example, p. 5 (B) Determine the components of the hiker s resultant displacement R for the trip. Find an expression for R in terms of unit vectors. Solution: The resultant displacement for the trip R = A + B has components given by Eqs. 3-10 & 3-11 (p. 44): R x = A x + B x = 17.7 km + 20.0 km = 37.7 km R y = A y + B y = -17.7 km + 34.6 km = 16.9 km In unit-vector form, we can write the total displacement as R = (37.7 î + 16.9 ĵ) km 39
hiking example, p. 6 Using Eq. 3-6 (p. 42), we find that vector R has a magnitude = 41.3 km & points 24.1 N of E. Now finalize. Units of R are km, which is reasonable for a displacement. From graphical representation in figure, estimate that hiker s final position ~ (38 km, 17 km), consistent with components of R in final result. Also, both components of R > 0, putting final position in 1st quadrant of coordinate system, also consistent with figure. 40
Problem-solving strategy: Adding vectors Select a coordinate system Try to select a system that minimizes # of components you must deal with Draw a sketch of vectors & label each 41
Problem-solving strategy: Adding vectors, 2 Find x & y components of each vector & x & y components of resultant vector Find z components if necessary Use Pythagorean theorem to get resultant s magnitude & tangent function to get its direction Other appropriate trig functions may be needed 42