To find the vector $\vec{u} = \overrightarrow{AB}$ given points $A(1, 3, -6)$ and $B(5, 0, 4)$, we use the formula:

$\overrightarrow{AB} = B - A = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$

Substitute $A = (1, 3, -6)$ and $B = (5, 0, 4)$:

1. Calculate the x-component: $x_2 - x_1 = 5 - 1 = 4$
2. Calculate the y-component: $y_2 - y_1 = 0 - 3 = -3$
3. Calculate the z-component: $z_2 - z_1 = 4 - (-6) = 4 + 6 = 10$

So, the vector $\vec{u} = \overrightarrow{AB}$ is:

$\vec{u} = \langle 4, -3, 10 \rangle$

Answer:

$\vec{u} = \langle 4, -3, 10 \rangle$

Would you like further details, or have any questions?

Related Questions:

1. How do you find the magnitude of vector $\vec{u}$?
2. What is the direction angle of $\vec{u}$ relative to the x-axis?
3. How can we find a unit vector in the direction of $\vec{u}$?
4. How do you calculate the midpoint between points $A$ and $B$?
5. How can we determine if another point $C$ is collinear with $A$ and $B$?

Tip:

To find the direction of a vector, divide it by its magnitude to get the unit vector in the same direction.