Problem 67
Question
Finding the Component Form of a Vector In Exercises \(67-74\) , find the component form of \(v\) given its magnitude and the angle it makes with the positive \(x\) -axis. Sketch y. $$\begin{array}{ll}{\text { Magnitude }} & {\text { Angle }} \\\ {\|\mathbf{v}\|=3} & {\theta=0^{\circ}}\end{array}$$
Step-by-Step Solution
Verified Answer
The component form of the vector v is \(v = 3i + 0j\).
1Step 1: Identify the magnitude and angle
For this problem, the magnitude, denoted as \(\| \mathbf{v} \|\), is given as 3. The angle \(\theta\) that vector v makes with the positive x-axis is given as \(0^{\circ}\).
2Step 2: Convert the angle to radians
Since trigonometric functions in the vector component formula expect the angle in radians, convert \(0^{\circ}\) to radians: \(\theta = 0^{\circ} * \frac{\pi}{180^{\circ}} = 0\) rad.
3Step 3: Calculate the x component
The x component of the vector is calculated by multiplying the magnitude by the cosine of the angle: \(v_x = \| \mathbf{v} \| * \cos(\theta) = 3 * \cos(0) = 3\).
4Step 4: Calculate the y component
The y component of the vector is calculated by multiplying the magnitude by the sine of the angle: \(v_y = \| \mathbf{v} \| * \sin(\theta) = 3 * \sin(0) = 0\).
Key Concepts
Vector MagnitudeTrigonometry in VectorsRadians and Degrees Conversion
Vector Magnitude
Understanding the magnitude of a vector is central to grasping vector operations in mathematics and physics. Essentially, the magnitude represents the length or the size of the vector. For instance, if we consider the vector \( \mathbf{v} \) as an arrow, the magnitude is the length of that arrow from its starting point to the end.
When working with the component form of a vector, we can visualize this as traveling along the x and y axis in a plane. The magnitude helps determine how far the vector extends along these axes. Mathematically, if a vector \( \mathbf{v} \) is defined by its components as \( \mathbf{v} = (v_x, v_y) \) then its magnitude can be calculated using the Pythagorean theorem as \( \|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2} \).
In our exercise, the vector's magnitude was given directly, which greatly simplifies finding its components. This is a common starting point in physics problems involving force or velocity vectors, where you typically know how strong the force is or how fast an object is moving before analyzing its directional components.
When working with the component form of a vector, we can visualize this as traveling along the x and y axis in a plane. The magnitude helps determine how far the vector extends along these axes. Mathematically, if a vector \( \mathbf{v} \) is defined by its components as \( \mathbf{v} = (v_x, v_y) \) then its magnitude can be calculated using the Pythagorean theorem as \( \|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2} \).
In our exercise, the vector's magnitude was given directly, which greatly simplifies finding its components. This is a common starting point in physics problems involving force or velocity vectors, where you typically know how strong the force is or how fast an object is moving before analyzing its directional components.
Trigonometry in Vectors
Another crucial concept in understanding vectors is the role of trigonometry. Specifically, the trigonometric functions sine and cosine are used to break a vector into its horizontal (x) and vertical (y) components.
The x component can be found by multiplying the magnitude of the vector by the cosine of the angle it forms with the positive x-axis: \( v_x = \| \mathbf{v} \| \cdot \cos(\theta) \). The y component is found using the sine function: \( v_y = \| \mathbf{v} \| \cdot \sin(\theta) \).
These equations assume that the vector's angle is measured from the positive x-axis, which is the standard in mathematics. Remember that cosine gives us the ratio of the adjacent side to the hypotenuse in a right triangle, which in vector terms translates to the horizontal stretch, while sine gives us the ratio of the opposite side to the hypotenuse, or the vertical stretch.
In the exercise, we utilized these trigonometric functions to decompose the vector into its components. The simplicity of this exercise is heightened by the angle \( \theta \) being \(0^\circ\), where \( \cos(0^\circ) \) is 1 and \( \sin(0^\circ) \) is 0, which leads to a straightforward computation and a better initial understanding of how trigonometry intertwines with vector operations.
The x component can be found by multiplying the magnitude of the vector by the cosine of the angle it forms with the positive x-axis: \( v_x = \| \mathbf{v} \| \cdot \cos(\theta) \). The y component is found using the sine function: \( v_y = \| \mathbf{v} \| \cdot \sin(\theta) \).
These equations assume that the vector's angle is measured from the positive x-axis, which is the standard in mathematics. Remember that cosine gives us the ratio of the adjacent side to the hypotenuse in a right triangle, which in vector terms translates to the horizontal stretch, while sine gives us the ratio of the opposite side to the hypotenuse, or the vertical stretch.
In the exercise, we utilized these trigonometric functions to decompose the vector into its components. The simplicity of this exercise is heightened by the angle \( \theta \) being \(0^\circ\), where \( \cos(0^\circ) \) is 1 and \( \sin(0^\circ) \) is 0, which leads to a straightforward computation and a better initial understanding of how trigonometry intertwines with vector operations.
Radians and Degrees Conversion
In trigonometry, angles can be expressed in degrees or radians. The relationship between these two units of measure is pivotal when dealing with vectors and trigonometric functions. Most mathematical functions, especially those within calculators and programming languages, expect angles to be in radians.
To convert degrees to radians, we multiply the number of degrees by \( \frac{\pi}{180} \). Conversely, to convert radians to degrees, we multiply the number of radians by \( \frac{180}{\pi} \).
This conversion is necessary because radians are a 'natural' measure of angle based on the radius of a circle, making them inherently compatible with the trigonometric functions derived from the unit circle. Degrees, on the other hand, are more artificial and human-centric, divided as they are into 360 parts for historical reasons.
When converting in the initial problem, the angle given was \(0^\circ\), which is equal to \(0\) radians, since \(0^\circ \times \frac{\pi}{180^\circ} = 0\). Such conversions are routine in problems dealing with rotation or periodic phenomena, and mastering them can streamline the process of finding vector components significantly.
To convert degrees to radians, we multiply the number of degrees by \( \frac{\pi}{180} \). Conversely, to convert radians to degrees, we multiply the number of radians by \( \frac{180}{\pi} \).
This conversion is necessary because radians are a 'natural' measure of angle based on the radius of a circle, making them inherently compatible with the trigonometric functions derived from the unit circle. Degrees, on the other hand, are more artificial and human-centric, divided as they are into 360 parts for historical reasons.
When converting in the initial problem, the angle given was \(0^\circ\), which is equal to \(0\) radians, since \(0^\circ \times \frac{\pi}{180^\circ} = 0\). Such conversions are routine in problems dealing with rotation or periodic phenomena, and mastering them can streamline the process of finding vector components significantly.
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