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Unformatted text preview: through the z axis at
z0 . We represent P as the triple (x0 , y0 , z0 ).
The coordinate axes in 3dimensional space divide space into
8 octants determined by the eight possible sign combinations
of the three coordinates.
The octant consisting of points with all positive coordinates is
called the ﬁrst octant. Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vectors in Space
Denote Cartesian three space as R3 .
A vector v in R3 is a directed line
segment from an initial
pointP (x0 , y0 , z0 ) to a terminal
point Q (x1 , y1 , z1 ). z Similar to vectors in the plane, v is
represented with an ordered triple
v = v1 , v2 , v3 , where Q
P v1 = x1 − x0
v2 = y1 − y0
v3 = z1 − z0 .
The numbers v1 , v2 , and v3 are
called the components of v. x y Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vector Algebra by Components
Theorem
Suppose v = v1 , v2 , v3 and w = w1 , w2 , w3 , and let λ be a
scalar.
(a) v + w = v1 + w1 , v2 + w2 , v3 + w3 .
(b) λv = λv1 , λv2 , λv3 .
(c) v − w = v1 − w1 , v2 − w2 , v3 − w3 .
Example
1 Find the sum of the vectors v = 7, −3, 2 and w = −5, 8, 4 . 2 1
If v = 6, −4, 3 , then ﬁnd −5v and 2 v. 3 Find the components of v having initial point P (2, 4, −3) and
terminal point Q (7, 5, 4). Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Translation of Axes
Sometimes it is convenient to translate the coordinate axes to
obtain a new coordinate system with axes parallel to the
original system but with a new origin.
If the translated origin has coordinates O (p , q , r ) in the
original system, then a point P has coordinates P (x , y , z ) in
the original system and coordinates P (x , y , z ) in the
translated system.
There is a nice relationship between the two coordinates given
by constructing the position vector
−→
−
O P = x , y , z = x − p, y − q, z − r .
Since the vector is the same regardless of the coordinate
system, we must have:
x = x − p, y = y − q, z =z −r Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space Example Example
If an xyz coordinate system is translated so that the origin is
O (1, −3, 5), ﬁnd P (2, 0, 4) in the new coordinate system. The Norm Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Properties of the Norm Theorem
Suppose v and w are vectors, and λ is a scalar.
(a) λv = λ v .
(b) v + w ≤ v + w (the triangle inequality).
Example
1 Find the norm of v = −5, 3, 2 . 2 Use vectors to ﬁnd the distance between P (1, 3, 0) and
Q (−3, −4, 6). 3 Verify the Triangle Inequality for the vectors v = 4, −3 and
w = 5 12. Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Unit Vectors If a vector has length 1, then it is called a unit vector.
Set i = 1, 0, 0 and j = 0, 1, 0 , and k = 0, 0, 1 .
These unit vectors can be used to write any vector
v = v1 , v2 , v3 in a unique way as a linear combination of the
form:
v = v1 i + v2 j + v3 k....
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 Spring '14
 KenyonJ.Platt
 Linear Algebra, Algebra, Vectors, Cartesian Coordinate System

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