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Question 31721: Ok i'm not so worried about the sketches I just want confirmation about mainly part c and d.
thank you.(~denotes an arrow)
Adjacent sides of a triangle in R3 are formed by the vectors ~u = 3~i −2~j+~k and ~v = −~j+3~k.
(a) With the aid of a rough sketch (not to scale and without axes) show clearly why the (scalar) component of ~u in the direction of ~v is given by ~u~v/||~v||, and find this component.
(b) Find a unit vector in the direction of ~v.
(c) Find ~ p1, the vector projection of ~u on ~v.
(d) Find vector ~ p2 orthogonal to ~v so that ~u decomposes into the sum ~ p1 + ~ p2.
(e) Show and label ~ p1 and ~ p2 clearly on your rough sketch.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Adjacent sides of a triangle in R3 are formed by the vectors ~u = 3~i −2~j+~k and ~v = −~j+3~k.
(a) With the aid of a rough sketch (not to scale and without axes) show clearly why the (scalar) component of ~u in the direction of ~v is given by ~u~v/||~v||, and find this component.
(b) Find a unit vector in the direction of ~v.
(c) Find ~ p1, the vector projection of ~u on ~v.
(d) Find vector ~ p2 orthogonal to ~v so that ~u decomposes into the sum ~ p1 + ~ p2.
(e) Show and label ~ p1 and ~ p2 clearly on your rough sketch.
LET US DROP ~ AND ASSUME VECTORS AS APPROPRIATLY.
YOU WANT C&D ONLY..OK...
C.
U=3I-2J+K AND V=-J+3K
PROJECTION OF U ON V IS GIVEN BY (U.V)/|V|
= (3I-2J+K).(-J+3K)/|(-J+3K)|=(-2*-1+1*3)/SQRT.(-1^2+3^2)=5/SQRT.(10)
THIS THE SCALAR PROJECTION.IF VECTOR PROJECTION IS NEEDED THEN
P1={5/SQRT.(10)}*UNIT VECTOR ALONG V =(5/SQRT.10)*(-J+3K)/SQRT.(10)
=(5/10)(-J+3K)=-J/2+3K/2
-------------------------------------------------------------
D.WE HAVE TO DECOMPOSE U INTO 2 VECTORS P1 AND P2 SO THAT P1+P2=U WHERE P1 IS ALONG V AND P2 IS ORTHOGONAL TO V.
WE ALREADY GOT P1...SO WE NEED TO FIND P2.
LET Y1=V...AND
Y2=U+AY1..WHERE Y2 IS ORTHOGONAL TO Y1.....DOTTING WITH Y1 WE GET
Y2.Y1=U.Y1+AY1.Y1.....SINCE Y2 AND Y1 ARE ORTHOGONAL,WE HAVE
0=U.Y1+AY1.Y1....OR......A=-U.V/(V.V)
A=-5/10=-1/2
HENCE WE HAVE
Y2=(3I-2J+K)-(1/2)(-J+3K)=(3I)-(3J/2)-(1K/2)
HENCE WE HAVE
3I-3J/2-K/2=(3I-2J+K)-(1/2)(-J+3K)
U=(3I-2J+K)=(3I-3J/2-K/2)+(1/2)(-J+3K)
P1=-J/2+3K/2
..AND
.P2=3I-3J/2-K/2
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