Check linear independence/dependence of vectors $a = m-n+p$, $b = m+n$, $c = n+\frac{1}{2} p$ and $d = n-p$. $m$, $n$, $p$ are not coplanar.

Question

DONE!!

Check linear independence/dependence of vectors

$a = m-n+p$,

$b = m+n$,

$c = n+(1/2)p$ and

$d = n-p.$

m, n, p are not coplanar.

I tried this: $xa+yb+zc+rd = 0$

$x(m-n+p)+y(m+n)+z(n+(1/2)p)+r(n-p)=0$

$m(x+y)+n(-x+y+z+r)+p(x+(1/2)z-r) = 0$

$x+y = 0$

$-x+y+z+r=0$

$x+(1/2)z-r = 0$

I get that they're not independent but i should get result $3a-3b+2c+4d=0$

Answer 1

If your three vectors $m,n,p$ are not coplanar, they form a basis of the three dimensional space $\operatorname{Span}(m,n,p)$, which is a three dimensional space. You have four vectors, $a,b,c,d$ in this space, they can't be independent since the space is only three-dimensional! You can then deduce that they are not independent.