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(MATH100)[2010](f)midterm~2283^_10414.pdf
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Math100, 3-Dimensional Space; Vectors
1.1 Rectangular Coordinates
Points in the space are presented by their coordinates, similar to the case for points in a plane. Take three lines in the space which are mutually perpendicular and we call them the x-axis, y-axis and the z-axis. The intersection of these three lines is called the origin O.
The xy-plane is the plane containing both the x-axis and the y-axis, and its similar for the yz-plane and the xz-plane.
The .rst (x), second (y), third (z) coordinates of a point P is the perpendic-ular distance from P the point to the yz-plane, xz-plane, xy-plane respectively.
For a function in one variable f , the graph of f is the set of points (x, y) in the plane satisfying y = f (x). The graph is a geometric object closely related to the function f.
In general, if f is a function in two variables. The graph of f is the set of points (x, y, z) in the space satisfying z = f (x, y). It is a surface in the space and is closely related to the study of f .
If (a, b, c) is a point in the space. The distance from the origin to this point
is a2 + b2 + c2 . This result is done by applying the Pythagoras theorem for twice. More generally, the distance between the points (a, b, c) and (., ., ) in
the space is (a . .)2 +(b . .)2 +(c . )2 .
22 2
Example 1.1 The set of points (x, y, z) in the space satisfying x+ y+ z=4 is the set of points having a distance 2 from the origin. Thus, it is a sphere with center the origin and radius 2.
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Example 1.2 The set of points (x, y, z) in the space satisfying y+ z=4 is the set of points having a distance 2 from the x-axis. Thus, it is a cylinder with its axis the x-axis and radius 2.
1.2 Vectors
De.nition 1.3 A vector is an ordered collection of numbers.
There are some operations de.ned for vectors:
De.nition 1.4 Addition:
(x1 , ..., xn)+(y1 , ..., yn)=(x1 + y1 , ..., xn + yn)
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De.nition 1.5 Scalar multiplication:
a(x1, ..., xn)=(ax1 , ...axn)
Theorem 1.6 For all vectors x, y, z and for all numbers a, b,
1.
(a + b)x = ax + bx
2.
(ab)x = a(bx)
3.
a(x + y)= ax + ay
4.
x + y = y + x
5.
(x + y)+ z = x +(y + z)
De.nition 1.7 Inner product (Dot product):
(x1 , ..., xn) (y1 , ..., yn)= x1y1 + ... + xnyn
Theorem 1.8 For all vectors x, y, z, and for all numbers a,
1.
(ax) y = a(x y)
2.
(x + y) z = x z + y z
3.
x y = y x
Vectors and their usual operations can be interpreted geometrically in the following way:
1.
A vector (x, y, z) is represented by a point in the space whose coordinates are x, y and z. It can also be represented by an arrow pointing from the origin to the point with coordinates x, y, z.
2.
The addition of two vectors u + v is the forth vertex of the parallelogram whose three other vertices are u, O and v.
3.
The scalar multiplication by a number a is to lengthen vectors by a factor
a.
De.nition 1.9 The length of a vecto