If a vector is a combination of two unit vectors, is this vector still a unit vector?
For example $\vec v_1$ and $\vec v_2$ are both unit vectors, that is, their lengths are both $1$.
Now if there is a vector $\vec f$.
$\vec f=\alpha \vec v_1+(1-\alpha )\vec v_2$ , $ 0<\alpha<1$
Is $\vec f$ also a unit vector?
On
We can check the conditions. So let $\vec{v}$ be a vector, being a lineair combination of 2 unit vectors $\vec{u}_1$ and $\vec{u}_2$. That is:
$\vec{v} = \alpha \vec{u}_1 + \beta \vec{u}_2$
Of course, here we suppose we are working in a 2D vectorspace spanned by the vectors $\vec{u}_1$ and $\vec{u}_2$. Taking the norm gives:
$||\vec{v}|| = \sqrt{\vec{v} \cdot \vec{v}} = \sqrt{\alpha^2 + \beta^2}$
The last gives the condition for $\vec{v}$ being a unit vector.
No. For instance, take:
$$(1,0),(0,1),\alpha=1/2$$
What is $\vec{f}$?
Note that $||\vec{f}||\neq 1$.
Is it clear? Good studies!