I've been studying linear algebra, then I've faced with this question:
Three vectors $p$, $q$, and $r$ sum to a zero vector and have the magnitude of $10$, $11$, and $15$, respectively. Determine the value of $p\cdot q + q\cdot r + r\cdot p$.
I came to the following conclusions:
p + q + r = 0 (obvious)
The following system of equations (obvious too):
xp + xq +xr = 0
yp + yq +yr = 0
zp + zq +zr = 0
And if I replace the dot products of the exercise, I have:
10.11.cos(teta1) + 11.15.cos(teta2) + 15.10.cos(teta3)
teta1 = angle between p and q;
teta2 = angle between q and r;
teta3 = angle between r and p.
It seems to me that the last equation is necessary, since the exercise supplies the module values. However, I don't see how to relate them to achieve a solution. Could you help me?
Easier: since$$(p+q+r)^2=(p+q+r)\cdot(p+q+r)=p^2+q^2+r^2+2(p\cdot q+q\cdot r+r\cdot p),$$ you want$$\tfrac12((p+q+r)^2-p^2-q^2-r^2)=-\frac12(10^2+11^2+15^2)=-223.$$