Section: 17 | Algebra |
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John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), CRC Press/Taylor & Francis, Boca Raton, FL.
If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

2 ALGEBRA

2.1 QUADRATIC FORMULA

The solutions of the equation ax2+bx+c=0 , where a0 , are given by: x=b±b24ac2a .

2.2 VECTOR ALGEBRA

2.2.1 Definitions

Any quantity which is completely determined by its magnitude is called a scalar. Examples include: mass, density, and temperature. Any quantity which is completely determined by its magnitude and direction is called a vector. Examples include: velocity, acceleration, force. A vector quantity is usually represented by a boldfaced letter such as V . Two vectors V1 and V2 are equal to one another if they have equal magnitudes and are acting in the same directions. A negative vector, written as V , is one which acts in the opposite direction to V , but is of equal magnitude to it. The magnitude of V is written |V| or simply v . The unit vectorV|V| (when |V|0) is that vector which has the same direction as V , but has a magnitude of unity (sometimes represented as V ).

The vector sum of V1 and V2 is represented by V1+V2 . The vector sum of V1 and V2 , or the difference of the vector V2 from V1 is represented by V1V2 .

If r is a scalar, then rV=Vr and this represents a vector r times the magnitude of V , in the same direction as V if r is positive, and in the opposite direction if r is negative. If r and s are scalars and V1 , V2 , V3 are vectors then the following rules of scalars and vectors hold:
V1+V2=V2+V1(r+s)V1=rV1+sV1r(V1+V2)=rV1+rV2V1+(V2+V3)=(V1+V2)+V3=V1+V2+V3
The vector 0 is a vector of zero length.

2.2.2 Vectors in Space

  1. A plane is described by two distinct vectors V1 and V2 . Should these vectors not intersect each other, then one can be displaced parallel to itself until they do. Any other vector V lying in this plane is given by
    V=rV1+sV2
  2. A position vector specifies the position in space of a point relative to a fixed origin. If V1 and V2 are the position vectors of the points A and B , relative to the origin O , then any point P on the line AB has a position vector V given by
    V=rV1+(1r)V2
    The scalar '' r '' can be taken as the metric representation of P since r=0 implies P=B and r=1 implies P=A . If the point P divides the line AB in the ratio r:s then
    V=(rr+s)V1+(sr+s)V2
  3. The vectors V1 , V2,V3,,Vn are said to be linearly dependent if there exist scalars r1,r2,r3,,rn , not all zero, such that
    r1V1+r2V2++rnVn=0
  4. A vector V is linearly dependent upon the set of vectors {V1,V2,V3,,Vn} if
    V=r1V1+r2V2+r3V3++rnVn
  5. Three vectors are linearly dependent if and only if they are co-planar.
  6. All points in space can be uniquely determined by linear dependence upon three base vectors i.e., three vectors any one of which is linearly independent of the other two. The simplest set of base vectors are the unit vectors along the coordinate axes. These are usually designated by i , j , and k .
  7. If V is a vector in space, and a , b , and c are the respective magnitudes of the projections of the vector along the axes then
    V=ai+bj+ck      and      |V|=a2+b2+c2
    and the direction cosines of V are
    cosα=a/v,   cosβ=b/v,   cosγ=c/v.
  8. The law of vector addition yields
    V1+V2=(a1+a2)i+(b1+b2)j+(c1+c2)k
  9. 2.2.3 The Scalar, Dot, or Inner Product of Two Vectors

    This product is represented as V1·V2 and is defined to be equal to v1v2cosθ , where v1=|V1|,v2=|V2| , and θ is the angle from V1 to V2 . That is
    V1·V2=v1v2cosθ=a1a2+b1b2+c1c2=V2·V1
    Note the relations:
    (V1+V2)·V3=V1·V3+V2·V3V1·(V2+V3)=V1·V2+V1·V3
    If V1 is perpendicular to V2 then V1·V2=0 , and if V1 is parallel to V2 then V1·V2=|V1||V2| . In particular:
    i·i=j·j=k·k=1i·j=j·k=k·i=0

    2.2.4 The Vector or Cross Product of Two Vectors

    This product is represented as V1×V2 and is defined as
    V1×V2=|V1||V2|sinθ1
    where θ is the angle from V1 to V2 and 1 is a unit vector perpendicular to the plane of V1 and V2 and so directed that a right-handed screw driven in the direction of 1 would carry V1 into V2 . Note that
    tanθ=|V1×V2|V1·V2
    The following rules apply to vector products:
    V1×V2=V2×V1V1×(V2+V3)=V1×V2+V1×V3(V1+V2)×V3=V1×V3+V2×V3V1×(V2×V3)=V2(V3·V1)V3(V1·V2)i×i=0,      j×j=0,      k×k=0i×j=k,      j×k=i,      k×i=j
    If V1=a1i+b1j+c1k , V2=a2i+b2j+c2k , and V3=a3i+b3j+c3k , then
    V1×V2=|ijka1b1c1a2b2c2|=(b1c2b2c1)i+(c1a2c2a1)j+(a1b2a2b1)k
    Note that, since V1×V2=V2×V1 , the vector product is not commutative.

    2.2.5 Scalar Triple Product

    There is only one possible interpretation of the expression V1·V2×V3 and that is V1·(V2×V3) which is a scalar. This product is called the scalar triple product and is written as [V1V2V3] . Further
    [V1V2V3]=V1(V2×V3)=(V1×V2)V3=V2(V3×V1)=|a1b1c1a2b2c2a3b3c3|=|V1||V2||V3|cosϕsinθ,
    where θ is the angle between V2 and V3 and φ is the angle between V1 and the normal to the plane of V2 and V3 . The determinant indicates that it can be considered as the volume of the parallelepiped whose three determining edges are V1 , V2 , and V3 . Note that cyclic permutation of the subscripts does not change the value of the scalar triple product: [V1V2V3]=[V2V3V1]=[V3V1V2] but [V1V2V3]=[V2V1V3] and [V1V1V2]0 .

    2.2.6 Vector Triple Product

    The product V1×(V2×V3) defines the vector triple product. The parentheses are vital to the definition.
    V1×(V2×V3)=(V1V3)V2(V1V2)V3=|ijka1b1c1|b2c2b3c3||c2a2c3a3||a2b2a3b3||
    This is a vector, perpendicular to V1 , lying in the plane of V2 and V3 . Similarly
    (V1×V2)×V3=|ijk|b1c1b2c2||c1a1c2a2||a1b1a2b2|a3b3c3|V1×(V2×V3)+V2×(V3×V1)+V3×(V1×V2)0
    If V1×(V2×V3)=(V1×V2)×V3 then V1,V2,V3 form an <i>orthogonal set</i>. Thus {i,j,k} form an orthogonal set.

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