https://file.aiquickdraw.com/mathbot/file/1730419219027gz2ddsr6.jpg

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?

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### 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.

Given the points A: (1, 3, -6) and B: (5, 0, 4), find the vector \( \vec{u} = \overrightarrow{AB} \).

Learn how to find the vector from point A(1, 3, -6) to point B(5, 0, 4) in 3D space. This solution includes step-by-step calculations, applying vector subtraction to determine \( \overrightarrow{AB} = \langle 4, -3, 10 \rangle \). Suitable for students in Grades 10-12.

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