Let's solve the problem.

The task is to compute the vector that originates from point $A(4, 5, 6)$ to point $B(3, -4, -7)$.

Formula for a vector between two points:

The vector $\vec{AB}$ is calculated as: $\vec{AB} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$ where $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$.

Substitution:

$\vec{AB} = \langle 3 - 4, -4 - 5, -7 - 6 \rangle$ $\vec{AB} = \langle -1, -9, -13 \rangle$

In component form:

The vector is: $\vec{AB} = -i - 9j - 13k$

Answer:

The correct option is D: $-i - 9j - 13k$.

Do you have any questions or want more details?

Related Questions:

1. How can vectors be visualized geometrically in 3D space?
2. What are the steps to calculate the magnitude of a vector?
3. What are some real-world applications of vectors?
4. How do you calculate the dot product of two vectors?
5. What is the cross product of two vectors, and when is it used?

Tip:

Always subtract the coordinates of the initial point from the terminal point when finding a vector!