Find a vector $\mathbf{a}$ with representation given by the directed line segment $\overrightarrow{AB}$: $A(0, 2, 5), \quad B(4, 2, -4)$.

Draw $\overrightarrow{AB}$ and the equivalent representation $\mathbf{a}$ starting at the origin.

Neetesh Kumar | December 18, 2024

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This is the solution to Math 1C
Assignment: 12.2 Question Number 3
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Step-by-step solution:

Step 1: Compute the vector $\overrightarrow{AB}$:

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

Given:

• $A(0, 2, 5)$,
• $B(4, 2, -4)$.

Substitute the coordinates into the formula:
$\overrightarrow{AB} = (4 - 0, 2 - 2, -4 - 5).$

Simplify each component:
$\overrightarrow{AB} = (4, 0, -9).$

Step 2: Represent the vector starting at the origin:

The vector $\mathbf{a}$ starting at the origin $(0, 0, 0)$ and ending at the same direction as $\overrightarrow{AB}$ is:
$\mathbf{a} = (4, 0, -9).$

1. The vector $\overrightarrow{AB}$ starts at:
   $(0, 2, 5)$ and ends at $(4, 2, -4)$.

2. The vector $\mathbf{a}$ starts at $(0, 0, 0)$ and ends at:
   $(4, 0, -9)$.

Final Answer:

1. Vector $\overrightarrow{AB}$:
   $\overrightarrow{AB} = \boxed{(4, 0, -9)}$

2. The vector $\overrightarrow{AB}$ starts at:
   $\boxed{(0, 2, 5)}$ and ends at $\boxed{(4, 2, -4)}$

3. The vector $\mathbf{a}$ starts at $(0, 0, 0)$ and ends at:
   $\boxed{(4, 0, -9)}$

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