Vector bisecting 2 vectors , out of which one is provided . Find the other vector and a unit vector in its direction.

Question

I was told that for a vector(a) bisecting two other vectors (b and c)

vector a= |b|vector c + |c|vector b

I couldnt understand what the book did as it didnt match with what i had learnt.

Answer 1

Given non-zero vectors $\vec{b}$ and $\vec{c}$ in $\mathbb{R}^3$, let us put $$ \vec{a} \colon= \lvert \vec{c} \rvert \vec{b} + \lvert \vec{b} \rvert \vec{c}. $$

Then we note that $$ \vec{a} - \vec{b} = \lvert \vec{c} \rvert \vec{b} + \lvert \vec{b} \rvert \vec{c} - \vec{b} = \big( \lvert \vec{c} \rvert - 1 \big) \vec{b} + \lvert \vec{b} \rvert \vec{c}. $$ And, $$ \vec{a} - \vec{c} = \lvert \vec{c} \rvert \vec{b} + \lvert \vec{b} \rvert \vec{c} - \vec{c} = \lvert \vec{c} \rvert \vec{b} + \big( \lvert \vec{b} \rvert - 1 \big) \vec{c}. $$

A visually impaired person, I'm unable to read the image you've incorporated into your post. Nor am I able to make sense of what you mean by a vector "bisecting" two other vectors.

Answer 2

If you divide through by $\mid b\mid\mid c\mid$ you get that $\lambda a=\frac b{\mid b\mid}+\frac c{\mid c\mid}$, where $\lambda=\frac 1{\mid b\mid\mid c\mid}$. (The RHS is the sum of the unit vectors in the directions of $b$ and $c$.)