7.5 Scalar and Vector Projections - La · PDF fileA Scalar Projection The scalar projection of...
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Calculus and Vectors – How to get an A+ 7.5 Scalar and Vector Projections ©2010 Iulia & Teodoru Gugoiu - Page 1 of 2 7.5 Scalar and Vector Projections A Scalar Projection The scalar projection of the vector a r onto the vector b r is a scalar defined as: θ cos || || ) ( a b onto a SProj r r r = where ) , ( b a r r ∠ = θ Ex 1. Given two vectors with the magnitudes 10 || || = a r and 16 || || = b r respectively, and the angle between them equal to ° = 120 θ , find the scalar projection a) of the vector a r onto the vector b r 5 120 cos 10 cos || || ) ( − = ° = = θ a b onto a SProj r r r b) of the vector b r onto the vector a r 8 120 cos 16 cos || || ) ( − = ° = = θ b a onto b SProj r r r B Special Cases Consider two vectors a r and b r . a) If b a r r ↑↑ ( 1 cos = θ ), then || || ) ( a b onto a SProj r r r = b) If b a r r ↑↓ ( 1 cos − = θ ), then || || ) ( a b onto a SProj r r r − = c) If b a r r ⊥ then 0 ) ( = b onto a SProj r r Ex 2. Find the scalar projection of the vector a r onto: a) itself || || ) ( a a onto a SProj r r r = b) the opposite vector a r − || || ) ( a a onto a SProj r r r − = − C Dot Product and Scalar Projection Recall that the dot product of the vectors a r and b r is defined as: θ cos || || || || b a b a r r r r = ⋅ So, the scalar projection of the vector a r onto the vector b r can be written as: || || cos || || ) ( b b a a b onto a SProj r r r r r r ⋅ = = θ Note: For a Cartesian (Rectangular) coordinate system, the scalar components x a , y a , and z a of a vector ) , , ( z y x a a a a = r are the scalar projections of the vector a r onto the unit vectors i r , j r , and k r . Proof: x z y x a a a a i i a i onto a SProj = ⋅ = ⋅ = 1 ) 0 , 0 , 1 ( ) , , ( || || ) ( r r r r r Ex 3. Given the vector ) 4 , 3 , 2 ( − = a r , find the scalar projection: a) of a r onto the unit vector i r 2 || || ) ( = ⋅ = i i a i onto a SProj r r r r r b) of a r onto the vector j i r r − 2 5 2 ) 0 , 1 , 1 ( ) 4 , 3 , 2 ( || || ) ( ) ( = − ⋅ − = − − ⋅ = − j i j i a j i onto a SProj r r r r r r r r c) of a r onto the vector k j i b r r r r + + − = 2 6 4 1 4 1 4 6 2 6 ) 1 , 2 , 1 ( ) 4 , 3 , 2 ( || || ) ( − = + + + − − = − ⋅ − = ⋅ = b b a b onto a SProj r r r r r d) of the unit vector i r onto the vector a r 29 2 16 9 4 ) 4 , 3 , 2 ( ) 0 , 0 , 1 ( || || ) ( = + + − ⋅ = ⋅ = a a i a onto i SProj r r r r r D Vector Projection The vector projection of the vector a r onto the vector b r is a vector defined as: || || cos || || ) ( b b a b onto a VProj r r r r r θ = E Dot Product and Vector Projection The vector projection of the vector a r onto the vector b r can be written using the dot product as: 2 || || ) ( ) ( b b b a b onto a VProj r r r r r r ⋅ = Note: For a Cartesian (Rectangular) coordinate system, the vector components i a a x x r r = , j a a y y r r = , and k a a z z r r = of a vector ) , , ( z y x a a a a = r are the vector projections of the vector a r onto the unit vectors i r , j r , and k r .
Transcript of 7.5 Scalar and Vector Projections - La · PDF fileA Scalar Projection The scalar projection of...
Calculus and Vectors – How to get an A+
7.5 Scalar and Vector Projections ©2010 Iulia & Teodoru Gugoiu - Page 1 of 2
7.5 Scalar and Vector Projections A Scalar Projection The scalar projection of the vector ar onto the vector b
r is a
scalar defined as: θcos||||)( abontoaSProj
rrr= where ),( ba
rr∠=θ
Ex 1. Given two vectors with the magnitudes 10|||| =a
r and 16|||| =br
respectively, and the angle between them equal to °=120θ , find the scalar projection a) of the vector ar onto the vector b
r
5120cos10cos||||)( −=°== θabontoaSProjrrr
b) of the vector b
r onto the vector ar
8120cos16cos||||)( −=°== θbaontobSProjrrr
B Special Cases Consider two vectors a
r and b
r.
a) If barr
↑↑ ( 1cos =θ ), then ||||)( abontoaSProjrrr
=
b) If barr
↑↓ ( 1cos −=θ ), then ||||)( abontoaSProjrrr
−=
c) If barr
⊥ then 0)( =bontoaSProjrr
Ex 2. Find the scalar projection of the vector ar
onto: a) itself
||||)( aaontoaSProjrrr
= b) the opposite vector ar−
||||)( aaontoaSProjrrr
−=−
C Dot Product and Scalar Projection Recall that the dot product of the vectors ar and b
r is defined
as: θcos|||||||| baba
rrrr=⋅
So, the scalar projection of the vector ar onto the vector br
can be written as:
||||cos||||)(
bbaabontoaSProj r
rrrrr ⋅
== θ
Note: For a Cartesian (Rectangular) coordinate system, the scalar components xa , ya , and za of a vector ),,( zyx aaaa =
r are
the scalar projections of the vector ar onto the unit vectors ir
, jr
, and kr
. Proof:
xzyx aaaa
iiaiontoaSProj =
⋅=
⋅=
1)0,0,1(),,(
||||)( r
rrrr
Ex 3. Given the vector )4,3,2( −=ar , find the scalar
projection: a) of ar onto the unit vector i
r
2||||
)( =⋅
=iiaiontoaSProj r
rrrr
b) of a
r onto the vector ji
rr−
25
2)0,1,1()4,3,2(
||||)()( =
−⋅−=
−−⋅
=−jijiajiontoaSProj rr
rrrrrr
c) of ar onto the vector kjib
rrrr++−= 2
64
141462
6)1,2,1()4,3,2(
||||)( −
=++
+−−=
−⋅−=
⋅=bbabontoaSProj r
rrrr
d) of the unit vector i
r onto the vector a
r
292
1694)4,3,2()0,0,1(
||||)( =
++
−⋅=
⋅=aaiaontoiSProj r
rrrr
D Vector Projection The vector projection of the vector ar onto the vector b
r is a
vector defined as:
||||cos||||)(
bbabontoaVProj r
rrrr
θ=
E Dot Product and Vector Projection The vector projection of the vector ar onto the vector br
can be written using the dot product as:
2||||)()(
bbbabontoaVProj r
rrrrr ⋅=
Note: For a Cartesian (Rectangular) coordinate system, the vector components iaa xx
rr= , jaa yy
rr= , and kaa zz
rr= of
a vector ),,( zyx aaaa =r are the vector projections of the
vector ar onto the unit vectors ir
, jr
, and kr
.
Calculus and Vectors – How to get an A+
7.5 Scalar and Vector Projections ©2010 Iulia & Teodoru Gugoiu - Page 2 of 2
Ex 4. Given two vectors )2,1,0( −=ar and )3,0,1(−=b
r, find:
a) the vector projection of the vector ar onto the vector b
r
⎟⎠⎞
⎜⎝⎛ −
=−
=++
−−=
⋅=
59,0,
53
10)18,0,6(
901)3,0,1)(6(
||||)()( 2bbbabontoaVProj r
rrrrr
b) the vector projection of the vector b
r onto the vector a
r
⎟⎠⎞
⎜⎝⎛ −
=−
=++−−
=⋅
=512,
56,0
5)12,6,0(
410)2,1,0)(6(
||||)()( 2aaabaontobVProj r
rrrrr
c) the vector projection of the vector a
r onto the unit vector kr
)2,0,0(1
)1,0,0)(2(||||)()( 2 −=
−=
⋅=
kkkakontoaVProj r
rrrrr
d) the vector projection of the vector i
r onto the vector ar
0)0,0,0(410)2,1,0)(0(
||||)()( 2
rr
rrrrr
==++−
=⋅
=aaaiaontoiVProj
Ex 5. Find an expression using the dot product of the vector components of the vector a
r along to the vector br
and along to a direction perpendicular to the direction of the vector b
rbut in the same plan containing the
vectors ar and b
r.
Let ba r
r|| be the vector component parallel to b
r. Then:
2|| ||||)()(
bbbabontoaVProja b r
rrrrrrr
⋅==
Let ba rr⊥ be the vector component perpendicular to b
r.
Then:
2|| ||||)(
bbbaaaaa bb r
rrrrrrr
rr⋅
−=−=⊥
Reading: Nelson Textbook, Pages 390-398 Homework: Nelson Textbook: Page 399 # 6, 11, 13, 14
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