Decomposing a Vector into Parts

[nycmathguy]

Section 6.4

Can you please do 58 as a guide for me to do the rest?

 

[MathLover1]

58.

given:

u=<0,3>
v=<2,15>

1: first find projection of u =<0,3>onto v =<2,15>

(u.v)/|v|*v/|v|

=(u*v)/(|v|)^2*v

u*v=0*2+3*15=45

(|v|)^2=(sqrt(2^2+15^2)^2=229

=45/229*<2,15>

=<2(45/229),15(45/229)>

=<90/229,675/229>

2: Find the orthogonal component

w2 = u - w1
w2 = <0,3> - <90/229,675/229>
w2 = <0-90/229,3-675/229>
w2 = <-90/229, 12/229>

3: Write the vector as the sum of two orthogonal vectors.

u = w1 + w2
u = <90/229,675/229> + <-90/229, 12/229>
u = <90/229-90/229, 675/229+12/229>
u =<0, 3>

 

[nycmathguy]

58.

given:

u=<0,3>
v=<2,15>

1: first find projection of u =<0,3>onto v =<2,15>

(u.v)/|v|*v/|v|

=(u*v)/(|v|)^2*v

u*v=0*2+3*15=45

(|v|)^2=(sqrt(2^2+15^2)^2=229

=45/229*<2,15>

=<2(45/229),15(45/229)>

=<90/229,675/229>

2: Find the orthogonal component

w2 = u - w1
w2 = <0,3> - <90/229,675/229>
w2 = <0-90/229,3-675/229>
w2 = <-90/229, 12/229>

3: Write the vector as the sum of two orthogonal vectors.

u = w1 + w2
u = <90/229,675/229> + <-90/229, 12/229>
u = <90/229-90/229, 675/229+12/229>
u =<0, 3>

What do you mean by w2 and w1?

 

[MathLover1]

the orthogonal components

 

[nycmathguy]

the orthogonal components

Thank you. I have my work cut out for me this weekend. Is this not a cool idea? You help me by answering questions.

I then use your step by step reply to answer similar textbook problems and only return here if needed. However, since I know you love mathematics and, just like me can't get enough numbers, I will keep you busy by posting David Cohen questions on the side. We both like David Cohen, right?

 

[nycmathguy]

the orthogonal components

 

[MathLover1]

correction:
u=<-56/17+5/17,-14/17-20/17>=<-3,-2>

 

[nycmathguy]

correction:
u=<-56/17+5/17,-14/17-20/17>=<-3,-2>

Ok. A basic math error that can happen to any of us.