# Sum of a vector parallel to b and a vector orthogonal to b

• Oct 15th 2008, 04:56 AM
Craka
Sum of a vector parallel to b and a vector orthogonal to b
Vectors are
$\displaystyle \begin{array}{l} \underline v = < 2, - 4 > \\ \underline b = < 1,1 > \\ \end{array}$

As far as a parallel vector to vector b went it would be just in the same direction and thus for expressing vector v as the sum of a vector parallel to vector b. I thought it would be
$\displaystyle \underline v + \underline b = \left\langle {(2 + 1)} \right.,\left. {( - 4 + 1)} \right\rangle = \left\langle 3 \right.,\left. { - 3} \right\rangle$

and for expressing the vector v as the sum of a vector orthogonal to vector b would be just
$\displaystyle \bot \underline {b = \left\langle { - 1,\left. 1 \right\rangle } \right.}$
and so
$\displaystyle \underline v + \bot \underline b = \left\langle {(2 + ( - 1)),\left. {( - 4 + 1)} \right\rangle = \left\langle {1,\left. { - 3} \right\rangle } \right.} \right.$

However the answer given for the orthogonal part of the question is $\displaystyle \left\langle { - 1,\left. { - 1} \right\rangle } \right.$
I'm not sure what I have done wrong in the later part of the question.
• Oct 15th 2008, 04:47 PM
Jhevon
Quote:

Originally Posted by Craka
Vectors are
$\displaystyle \begin{array}{l} \underline v = < 2, - 4 > \\ \underline b = < 1,1 > \\ \end{array}$

As far as a parallel vector to vector b went it would be just in the same direction and thus for expressing vector v as the sum of a vector parallel to vector b. I thought it would be
$\displaystyle \underline v + \underline b = \left\langle {(2 + 1)} \right.,\left. {( - 4 + 1)} \right\rangle = \left\langle 3 \right.,\left. { - 3} \right\rangle$

and for expressing the vector v as the sum of a vector orthogonal to vector b would be just
$\displaystyle \bot \underline {b = \left\langle { - 1,\left. 1 \right\rangle } \right.}$
and so
$\displaystyle \underline v + \bot \underline b = \left\langle {(2 + ( - 1)),\left. {( - 4 + 1)} \right\rangle = \left\langle {1,\left. { - 3} \right\rangle } \right.} \right.$

However the answer given for the orthogonal part of the question is $\displaystyle \left\langle { - 1,\left. { - 1} \right\rangle } \right.$
I'm not sure what I have done wrong in the later part of the question.

am i to understand that you are given vectors v and b and you want to express v as a sum of vectors parallel to b and then as a sum of vectors orthogonal to b?

if that is the case, why are you adding v to b and v to something parallel to b? it seems that you are not even attempting to do what the problem i think is asking. you want v = (sum of vectors parallel to b) and v = (sum of vectors orthogonal to b) right? how did you end up with v + b on the left side?
• Oct 15th 2008, 05:09 PM
Craka
Sorry the question states "Let v=<2,4> and b=<1,1>. Express v as the sum of a vector parallel to b and a vector orthogonal to b."
v and b above are vectors
• Oct 15th 2008, 05:21 PM
Jhevon
Quote:

Originally Posted by Craka
Sorry the question states "Let v=<2,4> and b=<1,1>. Express v as the sum of a vector parallel to b and a vector orthogonal to b."
v and b above are vectors

Hint: if $\displaystyle \bold{a}$ is a vector perpendicular to $\displaystyle \bold{b}$, then $\displaystyle \bold{a} \cdot \bold{b} = 0$

let $\displaystyle \bold{a} = \left< x,y \right>$. then we want $\displaystyle \left< x,y \right> \cdot \left< 1,1 \right> = 0$

also, any vectors parallel to $\displaystyle \bold{b}$, are of the form $\displaystyle n \bold{b}$. where $\displaystyle n$ is a scalar.

thus you want to write $\displaystyle \bold{v}$ in the following way

$\displaystyle \bold{v} = n \bold{b} + m \bold{a}$ for some scalars $\displaystyle m,n$

i suppose you can continue