Problem 28 - College Trigonometry

Question

Approximate the component form of the vector \(\vec{v}\) using the information given about its magnitude and direction. Round your approximations to two decimal places. \(\|\vec{v}\|=5280 ;\) when drawn in standard position \(\vec{v}\) makes a \(12^{\circ}\) angle with the positive \(x\) -axis

Step-by-Step Solution

Answer

The approximate component form of the vector is \(\langle 5164.57, 1097.71 \rangle\).

Step 1: Understanding Vector Components

A vector in standard position with a magnitude \( \|\vec{v}\| \) and an angle \( \theta \) with the positive \( x \)-axis can be expressed in terms of its components as \( \vec{v} = \langle v_x, v_y \rangle \), where \( v_x = \|\vec{v}\| \cos \theta \) and \( v_y = \|\vec{v}\| \sin \theta \). Here, the magnitude is 5280 and the angle is \(12^{\circ} \). This means the vector \( \vec{v} \) can be represented as \( \vec{v} = \langle 5280 \cos 12^{\circ}, 5280 \sin 12^{\circ} \rangle \).

Step 2: Calculating the x-component

Calculate \( v_x = 5280 \cos 12^{\circ} \). Use a calculator to find the cosine of \(12^{\circ}\) and multiply by 5280. \( \cos 12^{\circ} \approx 0.9781 \), so \( v_x \approx 5280 \times 0.9781 \approx 5164.57 \).

Step 3: Calculating the y-component

Calculate \( v_y = 5280 \sin 12^{\circ} \). Use a calculator to find the sine of \(12^{\circ}\) and multiply by 5280. \( \sin 12^{\circ} \approx 0.2079 \), so \( v_y \approx 5280 \times 0.2079 \approx 1097.71 \).

Step 4: Writing the Approximate Component Form

Now that we have both components, the approximate component form of \( \vec{v} \) is \( \langle 5164.57, 1097.71 \rangle \). Ensure that both component values are rounded to two decimal places.

Key Concepts

Understanding Magnitude and Direction of Vectors

Vectors are mathematical objects that have both a magnitude and a direction. The magnitude of a vector is a measure of its length or size, and it is always a non-negative value. It's denoted by \( \|\vec{v}\| \) and represents how long the vector is. Direction, on the other hand, tells us which way the vector is pointing in the coordinate system.
In our exercise, the vector has a magnitude of 5280, and it's pointing at an angle of \(12^{\circ} \) with respect to the positive \(x\)-axis. This angle is crucial because it helps us determine how the vector sits in orientation to our coordinate axes. Remember that the direction is often measured counter-clockwise from the positive \(x\)-axis in degrees or radians.

• Magnitude (\(\|\vec{v}\|\)): 5280
• Direction (angle \(\theta\)): 12° with the positive \(x\)-axis

Component Form of a Vector

The component form of a vector allows us to break down the vector into horizontal and vertical parts, making it easier to handle mathematically. For a vector \(\vec{v}\) in the plane, its component form is \( \langle v_x, v_y \rangle \). These components essentially measure how far and in which direction the vector extends along the \(x\)-axis (horizontal) and \(y\)-axis (vertical).
Using our exercise as an example, the vector can be decomposed into:

• \(v_x = \|\vec{v}\| \cos \theta \)
• \(v_y = \|\vec{v}\| \sin \theta \)

Substituting \(\|\vec{v}\| = 5280\) and \(\theta = 12^{\circ}\), we compute \(v_x\) and \(v_y\) to fully describe the vector in terms of its horizontal and vertical components, making them \( 5164.57 \) and \( 1097.71 \) respectively.

Trigonometric Functions in Vectors

Trigonometric functions such as sine and cosine are vital tools in vectors for relating angles to lengths and vice versa. When you have a vector with a known magnitude and direction angle, trigonometry allows you to calculate the precise horizontal and vertical components.
Consider the vector in this exercise, where you use the following formulas:

• Cosine function (\(\cos\)): Used to find the \(x\)-component. Represents how much of the vector's magnitude extends along the \(x\)-axis.
• Sine function (\(\sin\)): Used to find the \(y\)-component. Indicates how much of the vector falls along the \(y\)-axis.

These functions help translate a vector's overall movement into manageable parts by multiplying the magnitude by the cosine or sine of the angle \(\theta\). The accuracy hinges on knowing the angle and applying these trigonometric functions correctly.
In our exercise, \(\cos 12^{\circ} \approx 0.9781\) and \(\sin 12^{\circ} \approx 0.2079\), which when multiplied by the magnitude 5280, give the vector components \(v_x = 5164.57\) and \(v_y = 1097.71\) respectively.