Find all perp. vectors a, b w/ b is 3X length of a, & sum is

[onehotnight]

Find all perpendicular vectors a and b such that b is three times as long as a, and their sum is the vector [6,8]

I've started by coming up with two equations, but I can't seem to find two more in orde to complete linear algebra.

Ux+Vx = 6
Uy+Vy= 8

 

[Deleted member 4993]

Re: HELP!!!!

Find all perpendicular vectors a and b such that b is three times as long as a, and their sum is the vector [6,8]

I've started by coming up with two equations, but I can't seem to find two more in orde to complete linear algebra.

Ux+Vx = 6
Uy+Vy= 8

What does that condition tell you?

 

[onehotnight]

Re: HELP!!!!

dot product = 0

 

[Deleted member 4993]

Re: HELP!!!!

dot product = 0 <<< Use it

 

[soroban]

Hello, onehotnight!

You missed another condition . . .

Find all perpendicular vectors a and b such that b is three times as long as a
and their sum is the vector [6,8]

\(\displaystyle \text{Let the vectors be: }\:\begin{array}{ccc}\vec a &=& \langle p,q\rangle \\ \vec b &=& \langle r,s\rangle \end{array}\)

\(\displaystyle \vec a \perp \vec b \quad\Rightarrow\quad \langle p,q\rangle \cdot \langle r,s\rangle \:=\:0 \quad\Rightarrow\quad\boxed{ pr + qs \:=\:0}\)

\(\displaystyle |\vec b| \,=\,3|\vec a| \quad\Rightarrow\quad\sqrt{r^2+s^2} \:=\:3\sqrt{p^2+q^2} \quad\Rightarrow\quad\boxed{ r^2+s^2 \:=\:9(p^2+q^2)}\)

\(\displaystyle \vec a + \vec b \,=\,\langle6,8\rangle \quad\Rightarrow\quad\langle p,q\rangle + \langle r,s\rangle \:=\:\langle6,8\rangle \quad\Rightarrow\quad \boxed{\begin{array}{ccc}p+r &=& 6 \\ q+s &=& 8 \end{array}}\)