2/5/2003, VECTORS/DOT PRODUCT O. Knill, Math 21aHOMEWORK FOR FRIDAY: Section 9.2: 20, 34, Section 9.3: 18, 34, 38VECTORS. Two points P1= (x1, y1, z1), Q = P2= (x2, y2, z2) determine a vector ~v = (x2−x1, y2−y1, z2−z1).It points from P1to P2and we can write P1+ ~v = P2.COORDINATES. Points P in space are in one to one correspondence to vectors pointing from 0 to P . Thenumbers ~viin a vector ~v = (v1, v2, v3) are also called coordinates of the vector.REMARK: vectors can be drawn everywhere in space. If a vector starts at 0, then the vector ~v = (v1, v2, v3)points to the point (v1, v2, v3). That’s is why one can identify points P = (a, b, c) in space with a vector~v = (a, b, c). Two vectors which are translates of each other are considered equal.ADDITION SUBTRACTION, SCALAR MULTIPLICATION.xyuvu+vxyuvu-vxyu3 uBASIS VECTORS. The vectors~i = (1, 0, 0),~j = (0, 1, 0) and~k = (0, 0, 1) are called basis vectors.Every vector ~v = (v1, v2, v3) can be written as a sum of basis vectors: ~v = v1~i + v2~j + v3~k.WHERE DO VECTORS OCCUR?Velocity (see later): if (f(t), g(t))is a point in the plane whichdepends on time t, then ~v =(f0(t), g0(t)) is the velocity vec-tor at the point (f(t), g(t)).Forces: static problems involvethe determination of a force on ob-jects. Vectors appear also whendescribing fields like the electricfield or a wind velocity field.Fields: fields like elec-tromagnetic or gravita-tional fields or velocityfields in fluids are de-scribed with vectors.Qbits: in quantumcomputation, onedoes not work withbits, but with bits,which are vectors.Color Any color can be written asa vector ~v = (r, g, b), where r is thered component, g is the green com-ponent and b is the blue component.redgreenblue(r,g,b)Svg or Flash. Scalable Vector Graphics is an emerging standard for the web.It might rival soon Flash which is currently the most popular vector basedanimation tool. From www.w3.org: ”SVG is a language for describing two-dimensional graphics in XML. SVG allows for three types of graphic objects:vector graphic shapes (e.g., paths consisting of straight lines and curves), im-ages and text. Graphical objects can be grouped, styled, transformed andcomposited into previously rendered objects.VECTOR OPERATIONS: The ad-dition and scalar multiplication ofvectors satisfy ”obvious” properties:(no need memorizing them). Wewrite ∗ for multiplication with ascalar.~u + ~v = ~v + ~u commutativity~u + (~v + ~w) = (~u + ~v) + ~w additive associativity~u + 0 = ~u + 0 null vectorr ∗ (s ∗~v) = (r ∗ s) ∗~v scalar associativity(r + s)~v = ~v(r + s) distributivity in scalarr(~v + ~w) = r~v + r ~w distributivity in vector1 ∗~v = ~v one elementLENGTH. The length ||~v|| of ~v is the distance from the beginning to the end of the vector.EXAMPLEs. If ~v = (3, 4, 5) , then ||~v|| =√50 = 5√2.TRIANGLE INEQUALITY: ||~u + ~v|| ≤ ||~u|| + ||~v||.UNIT VECTOR. A vector of length 1 is called a unit vector. If ~v 6=~0, then ~v/||~v|| is a unit vector.EXAMPLE: If ~v = (3, 4), then ~v = (2/5, 3/5) is a unit vector,~i,~j,~k are unit vectors.PARALLEL VECTORS. Two vectors ~v and ~w are called parallel, if ~v = r ~w with some constant r.DOT PRODUCT. The dot product of two vectors ~v = (v1, v2, v3) and ~w = (w1, w2, w3) is defined as ~v · ~w =v1w1+ v2w2+ v3w3. Other notations are ~v · ~w = (~v, ~w) or < ~v|~w > (quantum mechanics) or viwi(Einsteinnotation) or gijviwj(general relativity). The dot product is also called scalar product, or inner product.LENGTH. Using the dot product one can express the length of ~v as ||~v|| =√~v ·~v.CHALLENGE. Express the dot product in terms of length only!SOLUTION: (~v + ~w, ~v + ~w) = (~v, ~v) + ( ~w, ~w) + 2(~v, ~w) can be solved for (~v, ~w).ANGLE. Because ||~v − ~w||2= (~v − ~w, ~v − ~w) = ||~v||2+ ||~w||2− 2(~v, ~w) isby the cos-theorem equal to ||~v||2+ ||~w||2− 2||~v|| · ||~w||cos(φ), where φis the angle between the vectors ~v and ~w, we have the important formula~v · ~w = ||~v|| · ||~w||cos(φ)FINDING ANGLES BETWEEN VECTORS. Find the angle between the vectors (1, 4, 3) and (−1, 2, 3).ANSWER: cos(φ) = 16/(√26√14) ∼ 0.839. So that φ = arccos(0.839..) ∼ 33◦.ORTHOGONALITY. Two vectors are called orthogonal if v · w = 0. The zero vector~0 is orthogonal to anyvector.PROJECTION. The vector ~a = ~w(~v · ~w/|~w|2) is called the projection of ~v onto ~w. The scalar projectionis defined as (~v · ~w/||~w||) . (Its absolute value is the length of the projection.)The vector~b = ~v −~a is the component of ~v orthogonal to the
View Full Document
Unlocking...