u – v
DESCRIPTION
|| ||. u. a 1 i + b 1 j. || ||. u – v. || ||. v. a 2 i + b 2 j. Theorem Angle between Vectors. y. 1. v. j. u. i. 1. x. Theorem. Two vectors v and w are orthogonal if and only if. - PowerPoint PPT PresentationTRANSCRIPT
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u
v
u – v
|| ||
|| ||
|| ||
a1 i + b1 j
a2 i + b2 j
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cos2
:Cosines of Law222
vuvuvu
cos2)()(
)()(2
22
22
12
1
221
221
vubaba
bbaa
cos222 2121 vubbaa
cos2121 vubbaa
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as
defined is thevectors,
twoare and If 2211
wvproductdot
jiwjiv
baba
2121 bbaa wv
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vvwv
jiwjiv
(b) (a)
find ,4 and 52 If
1542 (a) wv358
5522 (b) vv
4 25 29
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Theorem Angle between VectorsIf and are two nonzero vectors, the angle
, 0 < , between and is determined
by the formula
u v
u v
vu
vu
cos
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Find the angle between = 2 and
= 4 + 3 .
u i j
v i j
v u
vucos
5383142 vu
u 2 1 52 2
v 4 3 16 9 25 52 2
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5
1
55
5cos
v u
vu
4.63
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v i j 4 3
u i j 2
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cos ,sin
i
j
1
1
u
v
x
y
uvv
v v u
u i j cos sin
v v i j cos sin
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Theorem
Two vectors v and w are orthogonal if and only if
0wv
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Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal.
08124 wv
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w = 2i + 8j
v = 4i - jx
y