Problem 44 - Precalculus Mathematics for Calculus

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

Vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\) is \(-458.88 \mathbf{i} + 655.36 \mathbf{j}\).

Step 1: Understand the Problem

A vector with magnitude (length) 800 and angle (direction) 125 degrees is given. We need to find its horizontal (x-axis) and vertical (y-axis) components and express it in terms of Cartesian unit vectors \(\mathbf{i}\) and \(\mathbf{j}\).

Step 2: Calculate the Horizontal Component

The horizontal component can be found using the formula \(v_x = |\mathbf{v}| \cos \theta\). Substitute the given values: \(|\mathbf{v}| = 800\) and \(\theta = 125^{\circ}\). Thus, \(v_x = 800 \cos 125^{\circ}\).

Step 3: Calculate the Vertical Component

The vertical component is given by the formula \(v_y = |\mathbf{v}| \sin \theta\). Substitute the values: \(|\mathbf{v}| = 800\) and \(\theta = 125^{\circ}\). Thus, \(v_y = 800 \sin 125^{\circ}\).

Step 4: Compute \(\cos 125^{\circ}\) and \(\sin 125^{\circ}\)

Using a calculator, \(\cos 125^{\circ} = -0.5736\) and \(\sin 125^{\circ} = 0.8192\).

Step 5: Calculate Numerical Values for Components

Substitute \(\cos 125^{\circ}\) and \(\sin 125^{\circ}\) into the expressions for \(v_x\) and \(v_y\). \(v_x = 800 \times (-0.5736) = -458.88\) and \(v_y = 800 \times 0.8192 = 655.36\).

Step 6: Write the Vector in Terms of \(\mathbf{i}\) and \(\mathbf{j}\)

The vector \(\mathbf{v}\) can be written as \(\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j}\). Substituting the computed values, the vector is \(\mathbf{v} = -458.88 \mathbf{i} + 655.36 \mathbf{j}\).

Key Concepts

Horizontal Component

Let's dive into understanding the horizontal component of a vector. Imagine a vector as an arrow pointing in a specific direction. To figure out how much of this vector points horizontally (along the x-axis), we use the cosine function.

The horizontal component is calculated with the formula:

• \( v_x = |\mathbf{v}| \cos \theta \)

This means you multiply the length (or magnitude) of the vector by the cosine of the angle given. For example, if a vector's length is 800 and the angle is 125 degrees, you plug these into the formula:

• \( v_x = 800 \cos 125^{\circ} \)

Using a calculator, you find \( \cos 125^{\circ} = -0.5736 \). Multiply this by 800, giving you a horizontal component of -458.88.

This negative value indicates that the direction is to the left along the x-axis, which makes sense as 125 degrees points behind the vertical.

Vertical Component

Now, let's explore the vertical component, which represents the portion of the vector that points upward or downward (along the y-axis).

To find this component, the sine function comes in handy. The formula used is:

• \( v_y = |\mathbf{v}| \sin \theta \)

Using this, you take the vector's magnitude and multiply it by the sine of the angle to get the vertical part. Given our example, substituting the values yields:

• \( v_y = 800 \sin 125^{\circ} \)

After calculating, you find \( \sin 125^{\circ} = 0.8192 \), and multiply this by 800, resulting in a vertical component value of 655.36.

This positive value confirms that the vector has an upward component along the y-axis, aligning with our angle understanding, since 125 degrees points mostly upwards.

Unit Vectors

Lastly, unit vectors are our way of expressing vectors in a universal mathematical language, using \(\mathbf{i}\) and \(\mathbf{j}\).

These unit vectors are defined as directions with a length of one unit in Cartesian coordinates:

• \(\mathbf{i}\) is the unit vector in the horizontal (x-axis) direction.
• \(\mathbf{j}\) is the unit vector in the vertical (y-axis) direction.

In our case, we take the components we've calculated, \(v_x = -458.88\) and \(v_y = 655.36\), and express the vector as:

• \(\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j}\)
• \(\mathbf{v} = -458.88 \mathbf{i} + 655.36 \mathbf{j}\)

This brings a clear and precise representation of any vector in 2D space based on its horizontal and vertical components.

By writing vectors in terms of \(\mathbf{i}\) and \(\mathbf{j}\), it allows anyone to easily interpret and understand its direction and magnitude.