Vector Components

Understanding Vector Components

In studying vectors, we need to understand that every vector can be broken down into its components. Vector components are parts of a vector that are parallel to the coordinate axes.

Vector components are values that indicate how far a vector moves in the direction of the x-axis and y-axis. Every vector in a plane can be expressed as a linear combination of unit vectors $i$ and $j$ .

Vector Visualization and Its Components

Vector AB and its components on the x, y, and z axes.

If we have a vector $\overrightarrow{AB}$ , then:

\overrightarrow{AB} = a\cdot i + b\cdot j

where:

$a$ is the vector component on the x-axis (horizontal)
$b$ is the vector component on the y-axis (vertical)
$i$ is the unit vector in the direction of the x-axis
$j$ is the unit vector in the direction of the y-axis

Example of Vector Components

Consider the vector $\overrightarrow{AB}$ in the figure. This vector can be written as:

\overrightarrow{AB} = 6i + 8j

This means that vector $\overrightarrow{AB}$ has a horizontal component of 6 units to the right and a vertical component of 8 units upward.

Vector Magnitude from Its Components

When we know the components of a vector, we can calculate the length or magnitude of the vector using the Pythagorean theorem.

The magnitude of vector $\overrightarrow{AB}$ is denoted by $|\overrightarrow{AB}|$ and calculated using the formula:

|\overrightarrow{AB}| = \sqrt{a^2 + b^2}

where $a$ and $b$ are the components of the vector.

Example of Vector Magnitude Calculation

For the vector $\overrightarrow{AB} = 6i + 8j$ , its magnitude is:

|\overrightarrow{AB}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

Therefore, the magnitude of vector $\overrightarrow{AB}$ is 10 units.

Vectors in Three-Dimensional Space

Vectors are not limited to a plane (two dimensions) but can also be extended to three-dimensional space.

Vectors in 3D Space

Visualization of a vector and its components in three-dimensional space.

In three-dimensional space, a vector has three components: the x-component, y-component, and z-component. A vector in three-dimensional space can be expressed as:

\overrightarrow{v} = ai + bj + ck

where:

$a$ is the vector component on the x-axis
$b$ is the vector component on the y-axis
$c$ is the vector component on the z-axis
$i$ , $j$ , and $k$ are unit vectors in the direction of the x, y, and z axes

The magnitude of a vector in three-dimensional space is calculated using the formula:

|\overrightarrow{v}| = \sqrt{a^2 + b^2 + c^2}

Unit Direction Vector

To determine the direction of a vector, we can use a unit vector. A unit vector is a vector with a magnitude of 1 unit. To obtain a unit vector from a vector, we divide the vector by its magnitude.

The unit direction vector of $\overrightarrow{AB}$ is denoted by $\hat{AB}$ and calculated using:

\hat{AB} = \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|}

Example of Unit Direction Vector

For the vector $\overrightarrow{AB} = 6i + 8j$ with a magnitude of 10, its unit direction vector is:

\hat{AB} = \frac{6i + 8j}{10} = \frac{6}{10}i + \frac{8}{10}j = 0.6i + 0.8j

This unit vector indicates the direction of vector $\overrightarrow{AB}$ without regard to its magnitude.

Applications of Vector Components

Vector components have many applications in everyday life, such as:

Calculating velocity and displacement in physics
Analyzing forces in mechanics
Determining the direction and magnitude of resultants in object movement
Navigation and position determination in coordinate systems

By understanding vector components, we can analyze various problems involving direction and magnitude in mathematics and other applied sciences.

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