Question

Calculate a vector c in the same direction of the vector a and that has the same length as the vector b

Answer 1

For any scalar $t>0$, $t\vec a$ is a vector pointing in the same direction as $a$.

To force this vector to have the same length as another vector $\vec b$, first normalize $t\vec a$ by dividing it by its norm,

$\dfrac{t\vec a}{\|t\vec a\|}=\dfrac{t\vec a}{t\|\vec a\|}=\dfrac{\vec a}{\|\vec a\|}$

then multiply this vector by the norm of $\vec b$, so that

$\vec c=\dfrac{\|\vec b\|}{\|\vec a\|}\vec a$

To confirm that $\vec c$ has the same length as $\vec b$:

$\|\vec c\|=\left\|\dfrac{\|\vec b\|}{\|\vec a\|}\vec a\right\|=\dfrac{\|\vec b\|}{\|\vec a\|}\|\vec a\|=\|\vec b\|$