Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the component form of v given its magnitude and the angle it makes with the positive \(x\) -axis. Sketch v. Angle:\begin{aligned}&\theta=0^{\circ}\\\&\theta=45^{\circ}\\\&\theta=120^{\circ}\\\ &\theta=135^{\circ}\\\&\theta=150^{\circ}\\\&\theta=90^{\circ}\\\&\mathbf{v} \text { in the direction } \mathbf{i}+3 \mathbf{j}\\\&\mathbf{v} \text { in the direction } 3 \mathbf{i}+4 \mathbf{j} \end{aligned}. Magnitude:$$\|\mathbf{v}\|=3$$

Step-by-Step Solution

Answer

The vectors can be represented in the component form as (x, y) where x and y are the x and y components of the vector. To sketch them, plot these points on a Cartesian coordinate system and draw an arrow from the origin to the point.

Step 1: Calculating the x and y components of the vector

The x and y components of a vector can be calculated using the magnitude of the vector and the angle it makes with the x-axis (in radial form). The x component is calculated as \(\|v\|\cos(\theta)\) and the y component as \(\|v\|\sin(\theta)\). Here, \(\|v\|\) is the magnitude of the vector and \(\theta\) is the angle it makes with the positive x-axis.

Step 2: Converting angles from degrees to radians

Since trigonometric functions in mathematics usually work with angles in radians, convert the angle from degrees to radians before calculation. To convert degrees to radians, use the formula: rad = deg * \(\pi\) / 180.

Step 3: Plugging values into formula and calculation

After converting angles to radians, plug the values into the formula assigned for x and y component calculations. Repeat the same process for all the provided angles and calculate the components.

Step 4: Sketching the vector

Sketch each vector on the Cartesian coordinate system. Use the x component values on the x-axis and the y component values on the y-axis. Draw an arrow from the origin (0,0) to the point obtained from the component values.

Step 5: Vector Components for specific directions

For the given specific direction of vector v, i.e., along i + 3j and 3i + 4j, the components are simply the coefficients of i and j respectively. Therefore, the vector components for v in the direction i + 3j are (1,3) and in the direction 3i + 4j are (3,4).

Key Concepts

Magnitude and Direction

Understanding a vector's magnitude and direction is essential when dealing with vectors in trigonometry. The magnitude of a vector, often denoted as \( \|\mathbf{v}\| \), represents the length or strength of the vector in a coordinate system. Essentially, it tells you how far the vector travels from its origin point. Meanwhile, the direction of a vector is the angle it makes relative to the positive \( x \)-axis. This is often given in degrees or radians.When calculating the vector's properties, having both magnitude and direction allows you to determine its precise movement in space. For instance, in the example problem where the vector has a magnitude of 3, and angles such as \( 0^{\circ} \), \( 45^{\circ} \), or \( 135^{\circ} \) with the \( x \)-axis, you're able to fully describe its trajectory. Without both elements, a vector's position in the coordinate system cannot be accurately determined.

Component Form of Vectors

The component form of a vector is crucial when you want to break down the vector into its horizontal and vertical parts. This is especially important in physics and engineering, where each component can represent a different force or motion.For any vector \( \mathbf{v} \), if we know its magnitude and the angle \( \theta \) it makes with the positive \( x \)-axis, the component form can be found using trigonometry. The formula uses cosine for the \( x \)-component and sine for the \( y \)-component:

• \( x = \|\mathbf{v}\| \cos(\theta) \)
• \( y = \|\mathbf{v}\| \sin(\theta) \)

This translates the vector into the form \((x, y)\), which tells you the vector's movement along the \( x \)-axis and \( y \)-axis respectively. This is helpful, for example, when you need to display vector \( \mathbf{v} \) mathematically in the direction \( \mathbf{i} + 3\mathbf{j} \) as \((1, 3)\).

Trigonometric Functions and Radians

When dealing with vectors and angles, trigonometric functions such as cosine and sine are used extensively. Trigonometry helps relate the angles to the sides of a right triangle, which is fundamentally how vectors are broken down into components.However, these functions traditionally operate using radians rather than degrees. Hence, converting degrees into radians is a necessary step in the process. The conversion formula is simple: multiply degrees by \( \pi/180 \). For example, converting \( 45^{\circ} \) involves calculating \( 45 \times \pi/180 \), yielding \( \pi/4 \) radians.Using radians simplifies many mathematical formulas and makes calculations consistent with most scientific calculators and programming languages, ultimately offering more precision.

Sketching Vectors on Cartesian Plane

Once you've determined the component form of a vector, sketching it on the Cartesian plane becomes straightforward. This visual representation helps you see exactly where the vector points and how far it extends.To sketch a vector, start at the origin \((0,0)\), then plot the point \((x, y)\) using your previously calculated components. Draw a line from the origin to this point and finish with an arrowhead to indicate direction.For example, if your vector has components of \((1.5, 2.6)\), move right along the \( x \)-axis by 1.5 units and up along the \( y \)-axis by 2.6 units. This sketch allows you to grasp not just the direction but the magnitude visually, facilitating a better understanding of the concept overall.