Let $u$ and $v$ be two vectors in $R^2$ whose Euclidean norms satisfy $|u| = 2|v|$. What is the value $α$ such that $w = u + αv$ bisects the angle between $u$ and $v$?
Somewhere it explained as:
If we find two vectors with equal magnitude in the direction of given vectors, then their sum will bisect the angle between them.
So, in the vector $w = u + αv$
Hence $α = 2$
Can you explained this, please?
I am newbie with this, can you verify/explain this one line explanation, please.
Think about the parallelogram rule for adding vectors: their sum is the diagonal of the parallelogram with sides defined by the two vectors. The diagonals of a parallelogram bisect each other, so the midpoint of the ends of the vectors lies on this diagonal. In other words, if you look at the triangle formed by the origin and the ends of the two vectors, then the sum of the vectors bisects the side formed by those ends. If the two vectors have the same length, then you’ve got an isosceles triangle, in which case this side bisector also bisects the angle between the vectors.
So, to find an angle bisector of two vectors, you need to find two vectors that point in the same directions but also have equal lengths. Since $|u|=2|v|$, we know that $|2v|=|u|$, so $u+2v$ will bisect the angle.
More generally, for any two non-zero vectors $u$, $v$, an angle bisector is $|v|u+|u|v$.
$|u|=2|v|=|2v|$. So $u$ and $2v$ are vectors of the same length in directions of $u$ and $v$, so their sum $u+2v$ bisects the angle.