Problem 30 - Algebra and Trigonometry

Question

Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and j. $$ |\mathbf{v}|=800, \quad \theta=125^{\circ} $$

Step-by-Step Solution

Answer

The vector is \( -458.88 \mathbf{i} + 655.36 \mathbf{j} \).

Step 1: Understand the Components of a Vector

A vector can be split into horizontal and vertical components using the trigonometric functions cosine and sine. These components relate to the angle (θ) with the horizontal axis and the magnitude of the vector.

Step 2: Calculate the Horizontal Component

The horizontal component of a vector can be calculated as: \[ v_x = |\mathbf{v}|\cos(\theta) \]Given that \( |\mathbf{v}| = 800 \) and \(\theta = 125^{\circ} \), we find:\[ v_x = 800\cos(125^{\circ}) \]Using a calculator, \(\cos(125^{\circ}) \approx -0.5736 \):\[ v_x \approx 800 \times (-0.5736) = -458.88 \]

Step 3: Calculate the Vertical Component

The vertical component of a vector can be calculated as: \[ v_y = |\mathbf{v}|\sin(\theta) \]Using the given values:\( |\mathbf{v}| = 800 \) and \(\theta = 125^{\circ} \), we find:\[ v_y = 800\sin(125^{\circ}) \]Using a calculator, \(\sin(125^{\circ}) \approx 0.8192 \):\[ v_y \approx 800 \times 0.8192 = 655.36 \]

Step 4: Express the Vector in Terms of \( \mathbf{i} \) and \( \mathbf{j} \)

Once we have the horizontal and vertical components, we can write the vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \):\[\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \]Substituting the values found:\[\mathbf{v} = -458.88\mathbf{i} + 655.36\mathbf{j} \]

Key Concepts

Understanding Horizontal Components

To break down a vector into its essential parts, we first need to focus on the horizontal component. This component tells us how far the vector stretches along the horizontal (or x-axis) direction. Imagine splitting the vector into two lines: one straight across the horizontal axis and one upwards or downwards, depending on the angle.

The horizontal component, often represented as \( v_x \), is calculated using the cosine trigonometric function. This is because cosine relates the adjacent side of a right triangle to the hypotenuse. In our scenario, the adjacent side is the horizontal part, and the hypotenuse is the vector’s length.

• Formula: \( v_x = |\mathbf{v}|\cos(\theta) \)
• Where \( |\mathbf{v}| \) is the magnitude or length of the vector.
• \( \theta \) is the angle made with the horizontal axis.

For example, if your vector is 800 units long and inclined at 125 degrees, you would find the horizontal component by calculating \( 800\cos(125^{\circ}) \). This method enables us to express precisely what portion of the vector lies along the x-axis.

Exploring Vertical Components

The vertical component of a vector is equally crucial and indicates the vector's extent along the vertical (or y-axis) direction. While the horizontal component stretches across the plane, the vertical component shows motion upwards or downwards. Think of this as how much the vector moves directly north or south.

To compute the vertical component, \( v_y \), the sine function is used. Sine helps determine the relationship between the opposite side of a right-angled triangle and the hypotenuse. Here, the opposite side is the vertical length of our vector.

• Formula: \( v_y = |\mathbf{v}|\sin(\theta) \)
• This uses the vector's magnitude and the angle to detail the segment of the vector along the y-axis.

In practice, for a vector with a magnitude of 800 at an angle of 125 degrees, the vertical component would be \( 800\sin(125^{\circ}) \). This calculation clarifies how much the vector ascends or descends, relative to the starting point.

Decoding Trigonometric Functions

Trigonometric functions, namely sine and cosine, are indispensable tools in vector analysis. These functions allow us to dissect vectors into comprehensible parts that reveal detailed directions and magnitudes.

Sine (\( \sin \)) and Cosine (\( \cos \)) each play a specific role:

• Cosine: Used to find the horizontal component. It associates the adjacent side of an angle (here, the horizontal direction) with the hypotenuse (the vector itself).
• Sine: Used to determine the vertical component. It links the opposite side of the angle (here, the vertical direction) to the hypotenuse.

Understanding these functions equips you to deconstruct any vector by its angle and given magnitude. You can then rewrite vectors using the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which form the basis of this configuration. This translation into components enhances our grasp of vectors as it provides a clear picture of their direction and strength across a plane.