Hi everyone.
Does anybody kno how to create seven vectors in R^2 with integer components with norm 5 and from this list of vectors must be given two which are parallel and two which are orthogonal.?
Thx for any help.
Is this what you mean?
$\displaystyle \begin{array}{cccc} {\left\langle {3,4} \right\rangle } & {\left\langle { - 3, - 4} \right\rangle } & {\left\langle { - 3,4} \right\rangle } & {\left\langle {3, - 4} \right\rangle } \\ {\left\langle { - 4, - 3} \right\rangle } & {\left\langle {4,3} \right\rangle } & {\left\langle { - 4,3} \right\rangle } & {} \\\end{array}
$
To work out which ones are orthogonal (that's at 90 degrees to each other) o or parallel.
Take a pair of vectors, e.g. (1, 2) and (3, 4).
Multiply the 1st numbers or each pair together (that gives you 1 x 3 = 3) in this case.
Multiply the 2nd numbers together (that gives you 2 x 4 = 8).
Add them together (that's 3 + 8 = 11).
If that comes to zero, they're orthogonal. (Here they're not.)
e.g. (1, 2) and (-4, 2) are orthogonal because (1x(-4)) + (2 x 2) = -4 + 4 = 0.
Parallel's more complicated except when the vectors are the same length (i.e. have the same norm).
If, when you add the 2 bits you get after you've done the multiplication, you get the same as the square of the norm of one of them (or minus the square of one of them), they're parallel.
E.g. (3, 4) and (4, 3) both have norm 5 (square root of 3^2 + 4^2). So the square of their norm is 25.
But (3x4) + (4x3) is 24, so they're not parallel.
So (3, 4), (-3, -4) are parallel because you get (3 x -3) + (4 x -4) = -9 + -16 = -25 which is minus the square of the norm.