Consider the points A and B given and answer the following...

Consider the points $A$ and $B$ given and answer the following questions: a) Find $|\overrightarrow{A B}|$ b) Find the components of the vector $\mathbf{u}=\overrightarrow{A B}$ and sketch it in standard position. c) Write the vector $\mathbf{v}=\frac{1}{|\overrightarrow{A B}|} \cdot \mathbf{u}$ in component form. d) Find $|\mathbf{v}|$ e) Sketch the vector $\mathbf{v}$ and compare it to $\mathbf{u}$. $$A(2,-4) \text { and } B(2,1)$$

Consider the points $A$ and $B$ given and answer the following questions:
a) Find $|\overrightarrow{A B}|$
b) Find the components of the vector $\mathbf{u}=\overrightarrow{A B}$ and sketch it in standard position.
c) Write the vector $\mathbf{v}=\frac{1}{|\overrightarrow{A B}|} \cdot \mathbf{u}$ in component form.
d) Find $|\mathbf{v}|$
e) Sketch the vector $\mathbf{v}$ and compare it to $\mathbf{u}$.
$$A(2,-4) \text { and } B(2,1)$$

Key Concepts

Vector Components

This concept involves representing a vector in terms of its horizontal (x) and vertical (y) components. When a vector is defined by two points in a coordinate system, its components can be found by subtracting the coordinates of the initial point from the coordinates of the terminal point, providing a clear algebraic representation of the vector's direction and magnitude.

Magnitude of a Vector

The magnitude (or length) of a vector is a measure of how long the vector is and is calculated using the Pythagorean theorem. Given the vector components along the axes, the magnitude is obtained by taking the square root of the sum of the squares of these components, which quantifies the distance between the two points defining the vector.

Unit Vector

A unit vector is a vector that has a magnitude of one and indicates only the direction of the original vector. It is obtained by dividing the original vector by its magnitude. Unit vectors are helpful because they provide a standardized way of representing direction without scaling effects, making them useful for various applications in geometry and physics.

Standard Position Representation

Representing a vector in standard position involves placing its tail at the origin of the coordinate system so that its head's coordinates directly represent the vector's components. This positioning simplifies the process of graphing and analyzing vectors, as it provides a common reference point for comparing vectors and performing vector operations such as addition, scalar multiplication, and normalization.

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