The triple cross product of vectors a, b, and c is defined as the vector The triple cross product is coplanar to the vectors b and c; it can be expressed via b and c as follows: a×b×c =
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3 Vectors a and b are collinear if and only if a×b = 0 In particular, a×a = 0 and
a(a×b) = b(a×b) =0
4 (λa)×b = a×(λb) = λ(a×b) (associativity with respect to a scalar factor).
5 The cross product of basis vectors is
i×i = j×j = k×k =0, i×j = k, j×k = i, k×i = j.
6 If the vectors are given by their coordinates a = (a x , a y , a z ) and b = (b x , b y , b z), then
a×b =
a x a y a z
b x b y b z
= (a y b z – a z b y )i + (a z b x – a x b z )j + (a x b y – a y b x)k (4.5.3.6)
7 The area of the parallelogram spanned by vectors a and b is equal to
S=|a×b|= a y a z
b y b z
2+a x a z
b x b z
2+a x a y
b x b y
2 (4.5.3.7)
8 The area of the triangle spanned by vectors a and b is equal to
S = 1
2|a×b|=
1
2 a y a z
b y b z
2+a x a z
b x b z
2+a x a y
b x b y
2 (4.5.3.8)
Example 1 The moment with respect to the point O of a force F applied at a point M is the cross product
of the position vector −−→ OM by the force F; i.e., M = −−→ OM×F.
4.5.3-3 Conditions for vectors to be parallel or perpendicular
A vector a is collinear to a vector b if
b = λa or a×b =0 (4.5.3.9)
A vector a is perpendicular to a vector b if
Remark. In general, the condition a⋅b =0implies that the vectors a and b are perpendicular or one of
them is the zero vector The zero vector can be viewed to be perpendicular to any other vector.
4.5.3-4 Triple cross product
The triple cross product of vectors a, b, and c is defined as the vector
The triple cross product is coplanar to the vectors b and c; it can be expressed via b and c
as follows:
a×(b×c) = b⋅(a⋅c) – c⋅(a⋅b). (4.5.3.12)
Trang 24.5.3-5 Scalar triple product of three vectors.
The scalar triple product of vectors a, b, and c is defined as the scalar product of a by the
cross product of b and c:
Remark. The scalar triple product of three vectors a, b, and c is also denoted by abc.
Properties of scalar triple product:
1 [abc] = [bca] = [cab] = –[bac] = –[cba] = –[acb].
2 [(a + b)cd] = [acd] + [bcd] (distributivity with respect to addition of vectors) This
property holds for any number of summands
3 [λabc] = λ[abc] (associativity with respect to a scalar factor).
4 If the vectors are given by their coordinates a = (a x , a y , a z ), b = (b x , b y , b z), and
c = (c x , c y , c z), then
[abc] =
a x a y a z
b x b y b z
c x c y c z
. (4.5.3.14)
5 The scalar triple product [abc] is equal to the volume V of the parallelepiped spanned by
the vectors a, b, and c taken with the sign + if the vectors a, b, and c form a right-handed
trihedral and the sign – if the vectors form a left-handed trihedral,
6 Three nonzero vectors a, b, and c are coplanar if and only if [abc] =0 In this case, the
vectors a, b, and c are linearly dependent; they satisfy a relation of the form c = αa + βb.
4.5.4 Curves and Surfaces in Space
4.5.4-1 Methods for defining curves
A continuous curve in three-dimensional space is the set of points whose coordinates satisfy
a system of parametric equations (4.5.2.19) This method for defining a curve is referred
to as parametric The curve can also be defined by an equivalent system of equations
(4.5.2.18), i.e., described as the intersection of two surfaces (see Paragraph 4.5.4-2)
Remark 1 A curve may have more than one branch.
Remark 2. One can obtain the equation of the projection of the curve (4.5.2.19) onto the plane OXY by eliminating the variable x from equations (4.5.2.19)
4.5.4-2 Methods for defining surfaces
A continuous surface in three-dimensional space is the set of points whose coordinates
satisfy a system of parametric equations (4.5.2.17) This method for defining a surface is
referred to as parametric The surface can also be determined by an equation (4.5.2.16) or
z = f (x, y).
Remark 1 A surface may have more than one sheet.
Remark 2 The surfaces determined by the equations
F (x, y, z) =0 and λF (x, y, z) =0
coincide provided that the constant λ is nonzero.
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Remark 3. For any real number λ, the equation
F1(x, y, z) + λF2(x, y, z) =0
describes a surface passing through the line of intersection of the surfaces (4.5.2.18), provided that this line exists.
Remark 4 The equation
F1(x, y, z)⋅F2(x, y, z) =0
describes the surface that is formed by points of both surfaces in (4.5.2.18) and does not contain any other points.
4.6 Line and Plane in Space
4.6.1 Plane in Space
4.6.1-1 Equation of plane passing through point M0 and perpendicular to vector N.
A plane is a first-order algebraic surface In a Cartesian coordinate system, a plane is given
by a first-order equation
The equation of the plane passing through a point M0(x0, y0, z0) and perpendicularly to
a vector N = (A, B, C) has the form
A (x – x0) + B(y – y0) + C(z – z0) =0, or (r – r0)⋅N =0, (4.6.1.1)
where r and r0 are the position vectors of the point M (x, y, z) and M0(x0, y0, z0),
re-spectively (see Fig 4.34) The vector N is called a normal vector Its direction cosines
are
A2+ B2+ C2, cos β =
B
√
A2+ B2+ C2, cos γ =
C
√
A2+ B2+ C2. (4.6.1.2)
N
M
M x y z( , , )
0
Figure 4.34 Plane passing through a point M0and perpendicularly to a vector N.
Example 1 Let us write out the equation of the plane that passes through the point M0( 1 , 2 , 1 ) and is
perpendicular to the vector N = (3 , 2 , 3 ).
According to (4.6.1.1), the desired equation is 3(x –1 ) + 2(y –2 ) + 3(z –1 ) = 0 or 3x + 2y + 3z – 10 = 0
4.6.1-2 General equation of plane
The general (complete) equation of a plane has the form
Ax + By + Cz + D =0, or r⋅N + D =0 (4.6.1.3)
It follows from (4.6.1.1) that D = –Ax0 – By0– Cz0 If one of the coefficients in the
equation of a plane is zero, then the equation is said to be incomplete:
Trang 4Figure 4.35 Basis in plane.
1 For D = 0, the equation has the form Ax + By + Cz = 0and defines a plane passing through the origin
2 For A =0(respectively, B =0or C =0), the equation has the form By + Cz + D =0
and defines a plane parallel to the axis OX (respectively, OY or OZ).
3 For A = D =0(respectively, B = D =0or B = D =0), the equation has the form By+Cz =0
and defines a plane passing through the axis OX (respectively, OY or OZ).
4 For A = B =0(respectively, A = C =0or B = C =0), the equation has the form Cz+D =0
and defines a plane parallel to the plane OXY (respectively, OXZ or OY Z).
4.6.1-3 Parametric equation of plane
Each vector −−−→ M0M = r – r0lying in a plane (where r and r0 are the position vectors of the
points M and M0, respectively) can be represented as (see Fig 4.35)
−−−→
where R1= (l1, m1, n1) and R2= (l2, m2, n2) are two arbitrary noncollinear vectors lying in
the plane Obviously, these two vectors form a basis in this plane The parametric equation
of a plane passing through the point M0(x0, y0, z0) has the form
r = r0+ tR1+ sR2, or
x = x0+ tl1+ sl2,
y = y0+ tm1+ sm2,
z = z0+ tn1+ sn2
(4.6.1.5)
4.6.1-4 Intercept equation of plane
A plane Ax + By + Cz + D =0that is not parallel to the axis Ox (i.e., A≠ 0) meets this axis
at a (signed) distance a = –D/A from the origin (see Fig 4.36) The number a is called the
x -intercept of the plane Similarly, one defines the y-intercepts b = –D/B (for B ≠ 0) and
the z-intercept c = –D/C (for C ≠ 0) Then such a plane can be defined by the equation
x
a + y
b + z
which is called the intercept equation of the plane.
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a
X
Y
Z
c
Figure 4.36 A plane with intercept equation.
N N
γ β α
0
X
Y Z
Figure 4.37 A plane with normalized equation.
Remark 1 Equation (4.6.1.6) can be obtained as the equation of the plane passing through three given points.
Remark 2. A plane parallel to the axis OX but nonparallel to the other two axes is defined by the equation y/b + z/c =1, where b and c are the y- and z-intercepts of the plane A plane simultaneously parallel
to the axes OY and OZ can be represented in the form z/c =1
Example 2 Consider the plane given by the general equation2x + 3y– z +6 = 0 Let us rewrite it in intercept form.
The x-, y-, and z-intercepts of this plane are
2 = –3, b= –
D
3 = –2, and c = –
D
– 1 =6.
Thus the intercept equation of the plane reads
x
– 3+
y
– 2 +
z
6 =0.
4.6.1-5 Normalized equation of plane
The normalized equation of a plane has the form
r⋅N0– p =0, or x cos α + y cos β + z cos γ – p =0, (4.6.1.7)
where N0 = (cos α, cos β, cos γ) is a unit vector and p is the distance from the plane to the origin; here cos α, cos β, and cos γ are the direction cosines of the normal to the plane (see Fig 4.37) The numbers cos α, cos β, cos γ, and p can be expressed via the coefficients A, B, C as follows:
A2+ B2+ C2 , cos β =
B
√
A2+ B2+ C2 ,
A2+ B2+ C2 , p=
D
√
A2+ B2+ C2 ,
(4.6.1.8)
where the upper sign is taken if D <0and the lower sign is taken if D > 0 For D =0, either sign can be taken
The normalized equation (4.6.1.7) can be obtained from a general equation (4.6.1.3) by multiplication by the normalizing factor
√
where the sign of μ must be opposite to that of D.
Trang 6Example 3 Let us reduce the equation of the plane –2x+ 2y– z –6 = 0 to normalized form.
Since D = –6 < 0 , we see that the normalizing factor is
(– 2 )2+ 2 2 + (– 1 )2 =
1
3.
We multiply the equation by this factor and obtain
– 2
3x+
2
3y–
1
3z–2=0.
Hence for this plane we have
cos α = –2
3, cos β =
2
3, cos γ = –
1
3, p=2.
Remark. The numbers cos α, cos β, cos γ, and p are also called the polar parameters of a plane.
4.6.1-6 Equation of plane passing through point and parallel to another plane
The plane that passes through a point M1(x1, y1, z1) and is parallel to a plane Ax + By +
Cz + D =0is given by the equation
A (x – x1) + B(y – y1) + C(z – z1) + D =0 (4.6.1.10)
Example 4 Let us derive the equation of the plane that passes through the point M1( 1 , 2 , – 1 ) and is
parallel to the plane x +2y+ z +2 = 0
According to (4.6.1.1), the desired equation is (x –1 ) + 2(y –2) + (z +1 ) + 2 = 0 or
x+ 2y+ z –2 = 0
4.6.1-7 Equation of plane passing through three points
The plane passing through three points M1(x1, y1, z1), M2(x2, y2, z2), and M3(x3, y3, z3) (see Fig 4.38) is described by the equation
x – x1 y – y1 z – z1
x2– x1 y2– y1 z2– z1
x3– x1 y3– y1 z3– z1
=0, or
(r – r1)(r2– r1)(r3– r1)
=0, (4.6.1.11)
where r, r1, r2, and r3 are the position vectors of the points M (x, y, z), M1(x1, y1, z1),
M2(x2, y2, z2), and M3(x3, y3, z3), respectively.
M
M
M
M x y z( , , )
1
2
3
Figure 4.38 Plane passing through three points.
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Remark 1. Equation (4.6.1.11) means that the vectors −−−→ M1M , −−−→ M1M2, and −−−→ M1M3are coplanar.
Remark 2 Equation (4.6.1.11) of the plane passing through three given points can be represented via a fourth-order determinant as follows:
Remark 3. If the three points M1(x1, y1, z1), M2(x2, y2, z2), and M3(x3, y3, z3) are collinear, then
equa-tions (4.6.1.11) and (4.6.1.11a) become identities.
Example 5 Let us construct an equation of the plane passing through the three points M1( 1 , 1 , 1 ),
M2( 2 , 2 , 1), and M3( 1 , 2 , 2 ).
Obviously, the points M1, M2, and M3are not collinear, since the vectors −−−→ M1M2 = ( 1 , 1 , 0) and −−−→ M1M3 = ( 0 , 1 , 1 ) are not collinear According to (4.6.1.11), the desired equation is
=0,
whence
x – y + z –1 = 0
4.6.1-8 Equation of plane passing through two points and parallel to line
The equation of the plane passing through two points M1(x1, y1, z1) and M2(x2, y2, z2) and
parallel to a straight line with direction vector R = (l, m, n) (see Fig 4.39) is
x – x1 y – y1 z – z1
x2– x1 y2– y1 z2– z1
=0, or
(r – r1)(r2– r1)R
=0, (4.6.1.12)
where r, r1, and r2 are the position vectors of the points M (x, y, z), M1(x1, y1, z1), and
M2(x2, y2, z2), respectively.
R
R
M
M
M x y z( , , )
2
1
Figure 4.39 Plane passing through two points and parallel to line.
Remark. If the vectors −−−→ M1M2and R are collinear, then equations (4.6.1.12) become identities.
Example 6 Let us construct an equation of the plane passing through the points M1( 0 , 1 , 0) and M2( 1 , 1 , 1 )
and parallel to the straight line with direction vector R = (0 , 1 , 1 ).
According to (4.6.1.12), the desired equation is
1 – 0 1 – 1 1 – 0
=0,
whence
–x – y + z +1 = 0