MA261 Notes
Chapter 12: Vectors and the Geometry of Space
Section 12.1 Three-Dimensional Coordinate Systems
• x,y,z regular coordinates
o O(0,0,0)
o P(x,y,z)
√x +y +z 2
o |OP|=
• Example 1: Equation of a sphere with radius 2, centered at (1,-1, 1/2)
2
(x−1 + (y− −1 )+ z− 1 =4
o ( ) 2
2 2 2 7
o x −2x+ y +2y+z −z= 4
Section 12.2 Vectors
•
•
•
• •
• Example: a=<5,-12>, b=
o Find a + b
<5+-1,-12+-4>=<4,-16>
o Find 2a+3b
<2*5+2(-1),3(-12)+3(-4)>=<7,-36>
• Special Vectors
o i=<1,0,0>
o j=<0,1,0>
o k=<0,0,1>
Section 12.3 The Dot Product
•
•
•
• Example: a=i+2j-2k, b=4i-3k
o Find the angle between the vectors:
a dot b=1(4)+2(0)+(-2)(-3)=10
|a|=sqrt(1+2^2+(-2)^2)=3 |b|=sqrt(4^2+(-3)^2)=5
Theta=cos^(-1)(10/(3*5)=cos^(-1)(2/3)
• Two vectors a and b are orthogonal if and only if a dot b = 0
•
• Example: Use vectors to decide whether the triangle with vertices P(1,-3,-2),
Q(2,0,-4), and R(6,-2,-5) is right-angled.
o PQ=<1,3,-2>
o PR=<5,1,-3>
o PQ dot PR=(1)(5)+(3)(1)+(-2)(-3)=5+3+6=14, doesn’t = 0
o QP =
o QR = <4,-2,-1>
o QP dot QR = (-1)(4)+(-3)(-2)+(2)(-1)=-4+6-2=0
o So the angle at vertex Q is a right angle
Section 12.4 The Cross Product
•
• The vector a x b is orthogonal to both a and b
•
• Two nonzero vectors a and b are parallel if and only if a x b = 0 •
• The volume of the parallelepiped determined by the vectors a, b, and c is the
magnitude of their scalar triple product: V=|a dot (b x c)|
Section 12.5 Equations of Lines and Planes
r=r 0tv
• a vector equation of L
o Each value of parameter t gives the position vector r of a point on L
• Two vectors are equal if corresponding components are equal
o
o Called parametric equations
• Example: (a) Find a vector equation and parametric equations for the line that
passes through the point (5,1,3) and is parallel to the vector i+4j-2k. (b) Find two
other points on the line
r0=¿5,1,3>¿
o (a) =5i+j+3k and v=i+4j-2k
r=(5i+j+3k)+t(i+4j-2k)=(5+t)i+(1+4t)j+(3-2t)k
o (b) choosing the parameter value t=1 gives x=6,y=5,z=1 so (6,5,1) is a
point on the line. t=-1 gives the point (4,-3,5)
• Example: Find parametric equations and symmetric equations for the line. The
line through (5,1,0) and perpendicular to both i+j and j+k
o v=(i+j) x (j+k)=i-j+k
o parametric equations: x=5+t, y=1-t, z=t
o symmetric equations: x-5 = -(y-1)= z
• Symmetric equations
o
• Example: (a) Find parametric equations and symmetric equations of the line that
passes through the

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