Lecture 2: 6/25/2003, VECTORS/DOT PRODUCT O. Knill, Maths21aVECTORS. 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 components or of the vector.REMARKS: 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 u~u + ~v = (u1, u2, u3) + (v1, v2, v3)= (u1+ v1, u2+ v2, u3+ v3)~u −~v = (u1, u2, u3) − (v1, v2, v3)= (u1− v1, u2− v2, u3− v3)λ~u = λ(u1, u2, u3)= (λu1, λu2, λu3)BASIS VECTORS. The vectors~i = (1, 0, 0),~j = (0, 1, 0) and~k = (0, 0, 1) are called standard basis vectors.Every vector ~v = (v1, v2, v3) can be written as a sum of standard basis vectors: ~v = v1~i + v2~j + v3~k.WHERE DO VECTORS OCCUR? Here are some examples:Velocity (see later): if(f(t), g(t)) is a point inthe plane which dependson time t, then ~v =(f0(t), g0(t)) is the ve-locity vector at thepoint (f (t), g(t)).Forces: Some prob-lems in statics in-volve the determina-tion of a forces actingon objects. Forcesare represented asvectorsFields: 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 qbits,which are vectors.Color can be written asa vector ~v = (r, g, b),where r is red, g is greenand b is blue. An othercoordinate system is ~v =(c, m, y) = (1 − r, 1 −g, 1 −b), where c is cyan,m is magenta and y isyellow.redgreenblue(r,g,b)SVG. ScalableVector Graphics isan emerging stan-dard for the webfor describing two-dimensional graphicsin XML.VECTOR OPERATIONS: The ad-dition and scalar multiplication ofvectors satisfy ”obvious” properties.There is no need to memorize them.We write ∗ here for multiplicationwith a scalar but usually, the multi-plication sign is left out.~u + ~v = ~v + ~u commutativity~u + (~v + ~w) = (~u + ~v) + ~w additive associativity~u +~0 =~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 the one elementLENGTH. The length |~v| of ~v is the distance from the beginning to the end of the vector.EXAMPLES. 1) If ~v = (3, 4, 5) , then |~v| =√50 = 5√2. 2) |~i| = |~j| =~k| = 1, |~0| = 0.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+ v3w3Remark: in science, other notations are used: ~v · ~w = (~v, ~w) (mathematics) < ~v|~w > (quantum mechanics) viwi(Einsteinnotation) 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 the length alone.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) is by thecos-theorem equal to |~v|2+ |~w|2− 2|~v| · |~w|cos(α), where α is the anglebetween the vectors ~v and ~w, we get the important formula~v · ~w = |~v|· |~w|cos(α)CAUCHY-SCHWARZ INEQUALITY: |~v · ~w| ≤ |~v||~w| follows from that formula because |cos(α)| ≤ 1.TRIANGLE INEQUALITY: |~u + ~v| ≤ |~u| + |~v| follows from |~u + ~v|2= (~u + ~v) · (~u + ~v) = ~u2+ ~v2+ 2~u · ~v ≤~u2+ ~v2+ 2|~u ·~v| ≤ ~u2+ ~v2+ 2|~u|· |~v| = (|~u| + |~v|)2.FINDING ANGLES BETWEEN VECTORS. Find the angle between the vectors (1, 4, 3) and (−1, 2, 3).ANSWER: cos(α) = 16/(√26√14) ∼ 0.839. So that α = arccos(0.839..) ∼ 33◦.ORTHOGONAL VECTORS. Two vectors are called orthogonal if v · w = 0. The zero vector~0 is orthogonalto any vector. EXAMPLE: ~v = (2, 3) is orthogonal to ~w = (−3, 2).PROJECTION. The vector ~a = proj~w(~v) = ~w(~v · ~w/|~w|2) is called theprojection of ~v onto ~w.The scalar projection is defined ascomp~w(~v)= (~v · ~w)/|~w| . (Its ab-solute value is the length of the projection of ~v onto ~w.) The vector~b = ~v −~a is called the component of ~v orthogonal to the ~w-direction.vwabEXAMPLE. ~v = (0, −1, 1), ~w = (1, −1, 0), proj~w(~v) = (1/2, −1/2, 0), comp~w(~v) =
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