Problem 27 - 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. \(\|\vec{v}\|=63.92\); when drawn in standard position \(\vec{v}\) makes a \(78.3^{\circ}\) angle with the positive \(x\) -axis

Step-by-Step Solution

Answer

The component form of the vector \(\vec{v}\) is approximately \((13.29, 62.49)\).

Step 1: Understand the Problem

We need to find the component form of a vector \(\vec{v}\) given its magnitude \(\|\vec{v}\|=63.92\) and direction angle \(78.3^{\circ}\). The vector starts from the origin and forms an angle with the positive x-axis.

Step 2: Calculate the Vector Components

The component form of the vector \(\vec{v}\) can be expressed as \((x,y)\), where \(x\) is the horizontal component and \(y\) is the vertical component. These can be calculated using \(x = \|\vec{v}\| \cdot \cos(\theta)\) and \(y = \|\vec{v}\| \cdot \sin(\theta)\), where \(\theta = 78.3^{\circ}\).

Step 3: Use Cosine to Find the x-component

First, we calculate the x-component using \(x = \|\vec{v}\| \cdot \cos(78.3^{\circ})\). Using a calculator, \(\cos(78.3^{\circ}) \approx 0.2079\). So, \(x = 63.92 \cdot 0.2079 \approx 13.29\).

Step 4: Use Sine to Find the y-component

Next, we find the y-component using \(y = \|\vec{v}\| \cdot \sin(78.3^{\circ})\). Using a calculator, \(\sin(78.3^{\circ}) \approx 0.9781\). So, \(y = 63.92 \cdot 0.9781 \approx 62.49\).

Step 5: Write the Component Form

Now we can write the component form of \(\vec{v}\) as \((13.29, 62.49)\) where we rounded each component to two decimal places.

Key Concepts

Magnitude and Direction

Vectors have both magnitude and direction, distinguishing them from scalar quantities that have only magnitude. Imagine driving westward at a speed of 60 miles per hour. This is a velocity vector. It not only tells you how fast you're going, but also in which direction. In our specific problem, the magnitude of vector \(\vec{v}\) is given as 63.92, and it forms an angle of 78.3° with the positive x-axis.

The magnitude tells us how "long" the vector is. It's akin to asking "how powerful or fast," if you think about a vector like a force or speed. The direction, on the other hand, tells us precisely the vector's orientation relative to a defined line—in our case, the positive x-axis.

Understanding the vector's magnitude and direction is crucial for converting the vector into its component form, allowing us to break the vector down into its x (horizontal) and y (vertical) components.

Trigonometric Functions

Trigonometric functions like sine and cosine are invaluable in resolving vectors into components. These functions relate angles to side lengths in right-angled triangles, aiding us in understanding vector directions and their projections onto the coordinate axes.

Here's how they work for vector \(\vec{v}\):

• **Cosine Function (cos):** It helps us determine the adjacent side (or the x-component in standard position). We used \(x = \|\vec{v}\| \cdot \cos(78.3^{\circ})\). With the cosine of 78.3° approximately 0.2079, the calculation gives us the horizontal "reach" of the vector.
• **Sine Function (sin):** This is used for the opposite side (the y-component). So, \(y = \|\vec{v}\| \cdot \sin(78.3^{\circ})\) results in our vector's vertical "reach." With the sine of 78.3° approximately 0.9781, we calculate the y-component.

Trigonometry provides the mathematical tools to transform angles and magnitudes into precise coordinates on a plane, forming the backbone of vector analysis.

Vector Formulas

Vectors become much more manageable when broken down into components using predefined vector formulas. This method simplifies the understanding of vectors in geometry and physics by expressing them in an easily calculated format.

For vector \(\vec{v}\), with magnitude \(\|\vec{v}\|\) given and an angle \(\theta\) relative to the x-axis:

• **X-component (Horizontal) uses Cosine:** \[ x = \|\vec{v}\| \cdot \cos(\theta) \]
• **Y-component (Vertical) uses Sine:** \[ y = \|\vec{v}\| \cdot \sin(\theta) \]

This approach conveniently translates a vector into a two-dimensional plane, with each component representing a part of the overall vector in respective directions. The resulting vector, \((x, y)\), makes computations like additions, subtractions, and other vector operations straightforward and intuitive.

These standard formulas form a bridge from theoretical mathematics to practical applications, crucial for navigating fields like physics, engineering, and even computer graphics.