91Ó°ÊÓ

Unit vectors are the building blocks for representing directions in a plane or space. They have a magnitude of one and are used to indicate direction only.
- In two-dimensional space, the unit vectors commonly used are \( \hat{\mathbf{x}} \) and \( \hat{\mathbf{y}} \). - \( \hat{\mathbf{x}} \) represents the standard direction along the x-axis, while \( \hat{\mathbf{y}} \) represents the standard direction along the y-axis.
When describing a vector using these unit vectors, you're effectively breaking it down into its component parts: how much of it points in the x-direction and how much in the y-direction. This is particularly useful in physics and engineering to simplify complex vector problems by dealing with each direction separately.
In our problem, the displacement vector \( \Delta \overrightarrow{\mathbf{r}} \) is ultimately expressed as a combination of \( \hat{\mathbf{x}} \) and \( \hat{\mathbf{y}} \): \( 40.122\hat{\mathbf{x}} - 36.126\hat{\mathbf{y}} \). This shows how the particle moves across the x and y axes.

Trigonometric identities are mathematical tools that relate the angles and sides of triangles. They are essential in breaking down a vector into components along the axes.
For a vector making an angle \( \theta \) with the x-axis: - The cosine of the angle \( \cos(\theta) \) helps determine the x-component of the vector. - The sine of the angle \( \sin(\theta) \) helps determine the y-component of the vector.
In this exercise, the angle is 42 degrees below the x-axis. By recognizing that \( -42^{\circ} \) is equivalent in trigonometric calculations, we use: - \( \cos(-42^{\circ}) \) for the x-component calculation.- \( \sin(-42^{\circ}) \) for the y-component calculation.
This change of sign is because the trigonometric functions account for the direction in which the vector is pointing, helping accurately split the vector into its x and y parts.

The x-component of a vector is how much of that vector is aligned along the x-axis. Imagine only moving horizontally, that is what the x-component represents.
To find the x-component of \( \Delta \overrightarrow{\mathbf{r}} \), we use the cosine of the angle: \[ \Delta x = 54 \times \cos(-42^{\circ})\] This simplifies to approximately \( \Delta x = 40.122 \). This tells us that out of 54 meters, about 40.122 meters is the horizontal component of the particle's movement.
By using \( \hat{\mathbf{x}} \), we represent this horizontal displacement as \( 40.122\hat{\mathbf{x}} \), speaking to how far the vector goes in the direction of the x-axis.

The y-component measures how much of a vector is aligned along the y-axis. Think of this as representing vertical movement.
For the vector \( \Delta \overrightarrow{\mathbf{r}} \), the y-component is found using the sine function: \[ \Delta y = 54 \times \sin(-42^{\circ})\] This calculation results in approximately \( \Delta y = -36.126 \). The negative sign indicates that this portion of the vector is pointing downwards.
Thus, using the unit vector \( \hat{\mathbf{y}} \), the expression for this vertical component is \( -36.126\hat{\mathbf{y}} \), emphasizing the downward part of the displacement in the vector's movement.