Vectors (Chapter 11) 2898 Suppose a=2i¡j, b=i+3j, and c=¡ 4 i. Find:
a a+b b 3 b+c c a¡c
d 2 b¡a e jc+2aj f j¡ 2 bj9 Suppose a=μ
a 1
a 2¶
and b=μ
b 1
b 2¶. Prove that:
a if ka=b, k 6 =0, then a=
1
k
b b jkaj=jkjjaj10 Prove that ja+bj 6 jaj+jbj:
a using a geometric argument and the diagramb by letting a=μ
a 1
a 2¶
and b=μ
b 1
b 2¶In the diagram, point A has position vector¡!
OA=μ
a 1
a 2¶
,and point B has position vector¡!
OB=μ
b 1
b 2¶)¡!
AB=¡!
AO+¡!
OB
=¡
¡!
OA+
¡!
OB
=¡!
OB¡¡!
OA=μ
b 1
b 2¶
¡μ
a 1
a 2¶=μ
b 1 ¡a 1
b 2 ¡a 2¶Theposition vector of B relative to Ais¡!
AB=¡!
OB¡¡!
OA=μ
b 1 ¡a 1
b 2 ¡a 2¶In general, for two points A and B with position vectors
aandbrespectively, we observe
¡!
AB=¡a+b
=b¡a=μ
b 1 ¡a 1
b 2 ¡a 2¶and¡!
BA=¡b+a
=a¡b=μ
a 1 ¡b 1
a 2 ¡b 2¶D The vector between two points
abab+yxA(a 1 ,a_) 2B(b 1 ,b_) 2a 1 b 1a 2b 2OyxABa bO4037 Cambridge
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Y:\HAESE\CAM4037\CamAdd_11\289CamAdd_11.cdr Monday, 6 January 2014 1:03:24 PM BRIAN