Question

Find a vector a with representation given by the directed line segment ab?

Answer 1

The vector ( overrightarrow{a} ) from point ( a ) to point ( b ) represents the directed line segment ab ).

Explanation

A vector (overrightarrow{a} ) represents the displacement between two points, in this case, from ( a ) to ( b ). It indicates the direction and magnitude required to move from point ( a ) to point ( b ). To find ( overrightarrow{a} ), subtract the coordinates of a from the coordinates of ( b ) using vector subtraction.

This subtraction results in a vector that starts at point ( a ) and ends at point ( b ), depicting the direction and length of the directed line segment ( ab ).

Vectors are mathematical entities used to represent quantities that have both magnitude and direction. In geometry, vectors describe movements or displacements between points, providing a concise way to understand the relationship between two points in space.

Understanding vector operations like addition, subtraction, and multiplication allows for the determination of displacement vectors between different points in a coordinate system.

Answer 2

A vector a with representation given by the directed line segment ab is:

`Vector \( \mathbf{a} = \mathbf{b} - \mathbf{a} \).`

Step-by-step explanation:

To find a vector
`\( \mathbf{a} \)` with representation given by the directed line segment AB, we subtract the initial point vector (A) from the terminal point vector (B). This is based on the fact that a vector representing a displacement from one point to another can be obtained by subtracting the initial position vector from the final position vector. Mathematically,
`\( \mathbf{a} = \mathbf{b} - \mathbf{a} \).`

Let's break this down further. The vector
`\( \mathbf{b} \)`is the terminal point vector, representing the coordinates of point B. The vector
`\( \mathbf{a} \)` is the initial point vector, representing the coordinates of point A. Subtracting
`\( \mathbf{a} \) from \( \mathbf{b} \)` essentially gives us the vector that starts at the origin and ends at the coordinates of point B. This is precisely what we're looking for when representing the directed line segment AB as a vector.

In summary, the final answer
`\( \mathbf{a} = \mathbf{b} - \mathbf{a} \)`provides a vector that encapsulates both the direction and magnitude of the line segment AB, starting from the origin and pointing to the coordinates of point B.