Answer to Question #222515 in Analytic Geometry for Favvy1

Question

if a and b denote vectors oa and ob indicate on the same diagram the vectors oc and od denoted by a+b and a-b. draw on another diagram the vector oe denoted by a+2b obtain conditions that must be satisfied by the vectors a and b of each of the following conditions to hold
1.a+b bisect the angle between a and b
2.a+b=a-b

Answer 1

1) Since $\vec{OA}=\vec{BC}=\vec{a}$ and $\vec{AC}=\vec{OB}=\vec{b}$, OACB is a parallelogram, thus the angle ACO is equal to the angle COB. If $\vec{a}+\vec{b}$ (i.e. $\vec{OC}$) bisect the angle between $\vec{a}$ and $\vec{b}$, then the triangle OAC is isosceles, $OA=AC$. Thus, it must be $|\vec{a}|=|\vec{b}|$.

2) If $\vec{a}+\vec{b}=\vec{a}-\vec{b}$, then $\vec{b}=-\vec{b}$ or $2\vec{b}=\vec{0}$. Thus, in this case it must be $\vec{b}=\vec{0}$.