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u – v
12
u v u – v || || || || || || a 1 i + b 1 j a 2 i + b 2 j
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 Presentation
Transcript of u – v
u
v
u – v
|| ||
|| ||
|| ||
a1 i + b1 j
a2 i + b2 j
cos2
:Cosines of Law222
vuvuvu
cos2)()(
)()(2
22
22
12
1
221
221
vubaba
bbaa
cos222 2121 vubbaa
cos2121 vubbaa
as
defined is thevectors,
twoare and If 2211
wvproductdot
jiwjiv
baba
2121 bbaa wv
vvwv
jiwjiv
(b) (a)
find ,4 and 52 If
1542 (a) wv358
5522 (b) vv
4 25 29
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
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
5
1
55
5cos
v u
vu
4.63
v i j 4 3
u i j 2
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
Theorem
Two vectors v and w are orthogonal if and only if
0wv
Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal.
08124 wv
w = 2i + 8j
v = 4i - jx
y
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