Thread: Sum of a vector parallel to b and a vector orthogonal to b

1. Sum of a vector parallel to b and a vector orthogonal to b

Vectors are
$
\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
$
\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
$
\bot \underline {b = \left\langle { - 1,\left. 1 \right\rangle } \right.}
$

and so
$
\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 $
\left\langle { - 1,\left. { - 1} \right\rangle } \right.
$

I'm not sure what I have done wrong in the later part of the question.

2. Originally Posted by Craka
Vectors are
$
\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
$
\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
$
\bot \underline {b = \left\langle { - 1,\left. 1 \right\rangle } \right.}
$

and so
$
\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 $
\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?

3. 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

4. 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 $\bold{a}$ is a vector perpendicular to $\bold{b}$, then $\bold{a} \cdot \bold{b} = 0$

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

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

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

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

i suppose you can continue

expressing parallel and orthagonal vectors

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