Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. Direction angle \(115^{\circ},\) magnitude 0.6

Step-by-Step Solution

Answer

The vector is approximately -0.25i + 0.54j.

Step 1: Understand the Vector Components

A vector in the form \( ai + bj \) can be expressed using its magnitude and direction angle. The magnitude is given as 0.6 and the direction angle is 115°. To find \(a\) and \(b\), we will use trigonometry: \( a = \text{magnitude} \times \cos(\text{angle}) \) and \( b = \text{magnitude} \times \sin(\text{angle}) \).

Step 2: Calculate Component 'a' (i Component)

Calculate \( a \) using the cosine function. Since the angle is given in degrees, ensure your calculator is set to degree mode. \[ a = 0.6 \times \cos(115°) \approx 0.6 \times (-0.4226) = -0.2536 \]Round \(-0.2536\) to the nearest hundredth: \( a \approx -0.25 \).

Step 3: Calculate Component 'b' (j Component)

Calculate \( b \) using the sine function.\[ b = 0.6 \times \sin(115°) \approx 0.6 \times 0.9063 = 0.5438 \]Round \(0.5438\) to the nearest hundredth: \( b \approx 0.54 \).

Step 4: Write the Vector in the Form ai + bj

Combine \(a\) and \(b\) into the vector form, putting the calculated components together:\[ \vec{v} = -0.25i + 0.54j \].

Key Concepts

Direction Angle

When dealing with vectors, the direction angle is crucial. It specifies the direction in relation to a reference line, usually the positive x-axis. For example, a direction angle of 115° means the vector points 115° counterclockwise from the positive x-axis.
This angle helps in describing the orientation of the vector in a plane.

• A direction angle is often measured in degrees, but radians can also be used.
• We use direction angles to determine the individual components of a vector, which influences its representation and operations in coordinate systems.

Using direction angles in trigonometry allows for precise computations of components, ensuring we know how much the vector moves along the x and y axes.

Magnitude of Vectors

The magnitude of a vector is akin to its length or size. It shows how strong or significant a vector is, regardless of its direction. In the given problem, the magnitude is 0.6, indicating the length of the vector.

• Magnitude is always a non-negative number.
• It can be calculated from the vector components using the Pythagorean theorem, especially when given in rectangular form.
• A key aspect of the magnitude is that it scales the direction components, influencing both the vector's impact and direction in calculations.

Understanding magnitude is vital for converting between different forms of vector representation and analyzing vector behavior.

Trigonometry

Trigonometry plays a central role in vector calculations. It assists in breaking down vectors into components using their direction angles. For the exercise provided, we utilize trigonometric functions:

• Cosine (\(\cos\)) is used to find the horizontal component (\(a\) in \(ai + bj\)).
• Sine (\(\sin\)) determines the vertical component (\(b\) in \( ai + bj\)).

When a direction angle and the magnitude are known, these trigonometric functions offer a straightforward method to derive vector components:

• \[a = ext{magnitude} \times \cos(\text{angle})\]
• \[b = ext{magnitude} \times \sin(\text{angle})\]

Using trigonometry efficiently connects geometric angles to algebraic vector forms, making it indispensable in vector math.

Vector Components

Vector components are the fundamental parts of a vector when it is decomposed along the coordinate axes. In the expression \( ai + bj \), \(a\) and \(b\) represent the horizontal and vertical components, respectively.

• The horizontal component \(a\) is derived using the cosine of the direction angle.
• The vertical component \(b\) is obtained using the sine of the direction angle.

These components help express the vector in its rectangular form, making it easier to perform mathematical operations like addition or subtraction with other vectors.
By understanding vector components, we simplify complex vector operations and gain deeper clarity on how vectors behave within a coordinate system, facilitating various applications in physics and engineering.