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Unformatted text preview: e Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vector Algebra by Components
Theorem
Suppose v = v1 , v2 and w = w1 , w2 , and let λ be a scalar.
(a) v + w = v1 + w1 , v2 + w2 .
(b) λv = λv1 , λv2 .
(c) v − w = v1 − w1 , v2 − w2 .
Example
1 2 Find the sum of the vectors v = −5, 2 and w = 7, 1 using
componentwise addition, then verify that the parallelogram
law yields the same vector.
1
If v = −5, 2 , then ﬁnd 3 v and −3v. Describe the new
vectors. Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space Properties of Vector Algebra
Theorem
Suppose u, v, and w are vectors in the plane. Let λ and µ be
scalars.
(a) v + w = w + v.
(b) (v + w) + u = v + (w + u).
(c) v + 0 = v = 0 + v.
(d) v − v = 0.
(e) λ(µv) = (λµ)v.
(f) λ(v + w) = λv + λw.
(g) (λ + µ)v = λv + µv.
(h) 1v = v. The Norm Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Cartesian Coordinates of a Point in Space
We set up a Cartesian coordinate system in 3dimensional
space as follows.
Fix a plane and set up a Cartesian coordinate system in this
plane.
Call it the xy plane, with coordinate axes labeled x and y ,
respectively.
There is a line perpendicular to the xy plane passing through
the origin. Call this line the z axis.
Deﬁne the positive direction on the z axis as the direction
your thumb points if you curl the ﬁngers of your right hand
from the positive x axis toward the positive y axis through the
smallest angle. Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Cartesian Coordinates of a Point in Space
We have three Cartesian planes: the xy plane, the xz plane,
and the yz plane.
Any point P in space lies at the intersection of three planes:
One parallel to the yz plane and passing through the x axis at
x0 ;
One parallel to the xz plane and passing through the point y0
on the y axis;
One parallel to the xy plane and passing...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.
 Spring '14
 KenyonJ.Platt
 Linear Algebra, Algebra, Vectors, Cartesian Coordinate System

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