Find a vector $\vec{v}$ parallel to $\vec{u}=-5 \vec{i}-7 \vec{k}$ and with magnitude $\|\vec{v}\|=4$.
Enter your vector in terms of $\vec{i}, \vec{j}$ and $\vec{k}$. However, you only need to type the ordinary letters $i, j$, and $k$.
\[
\vec{v}=
\]

Step 1 ：Given vector \(\vec{u}=-5 \vec{i}-7 \vec{k}\) and we need to find a vector \(\vec{v}\) parallel to \(\vec{u}\) with magnitude \(\|\vec{v}\|=4\).

Step 2 ：First, we calculate the magnitude of \(\vec{u}\) using the formula \(\|\vec{u}\|=\sqrt{(-5)^2+(-7)^2}=8.60\).

Step 3 ：Next, we find the unit vector in the direction of \(\vec{u}\) by dividing \(\vec{u}\) by its magnitude. The unit vector \(\vec{u}_{unit}\) is \([-0.58, 0, -0.81]\).

Step 4 ：Finally, we find \(\vec{v}\) by multiplying the unit vector by the desired magnitude of \(\vec{v}\), which is 4. So, \(\vec{v}\) is \([-2.32, 0, -3.25]\).

Step 5 ：\(\boxed{\vec{v}=-2.32 \vec{i}-3.25 \vec{k}}\) is the vector parallel to \(\vec{u}=-5 \vec{i}-7 \vec{k}\) and with magnitude \(\|\vec{v}\|=4\).