Problem 72
Question
In Exercises 67-74, find the component form of \(\mathbf{v}\) given its magnitude and the angle it makes with the positive \(x\)-axis. Sketch \(\mathbf{v}\). Magnitude - ||\(\mathbf{v}\)|| \(= 4\sqrt{3}\) Angle - \(\theta = 90^{\circ}\)
Step-by-Step Solution
Verified Answer
The component form of the vector is \(\mathbf{v} = 0\mathbf{i} + 4\sqrt{3}\mathbf{j}\). In the sketch, the vector should be pointing straight up from the origin with a length of \(4\sqrt{3}\) units
1Step 1: Calculation of x-component
First calculate the x-component of the vector using the formula \(\mathbf{v}_x = ||\mathbf{v}||cos(\theta)\). Here, \(\mathbf{v}_x\) is the x-component of the vector, ||\mathbf{v}|| is the magnitude and \(\theta\) is the angle. Since the angle is \(90^{\circ}\), it needs to be converted into radians. \(\theta = 90^{\circ} = 90 * (\pi /180) = \pi /2\) radians. Hence, \(\mathbf{v}_x = 4\sqrt{3} * cos(\pi /2) = 4\sqrt{3} * 0 = 0\).
2Step 2: Calculation of y-component
Next, the y-component of the vector needs to be calculated using the formula \(\mathbf{v}_y = ||\mathbf{v}||sin(\theta)\). Here, \(\mathbf{v}_y\) is the y-component of the vector. Hence, \(\mathbf{v}_y = 4\sqrt{3} * sin(\pi /2) = 4\sqrt{3} * 1 = 4\sqrt{3}\).
3Step 3: Writing the component form and sketching the vector
The component form of the vector \(\mathbf{v}\) is \(\mathbf{v} = \mathbf{v}_x\mathbf{i} + \mathbf{v}_y\mathbf{j}\) = \(0\mathbf{i} + 4\sqrt{3}\mathbf{j}\). Thus, the vector \(\mathbf{v}\) lies entirely in the positive y-direction, with a length of \(4\sqrt{3}\). When sketching this, draw a line going straight up from the origin with a length of \(4\sqrt{3}\) units.
Key Concepts
Magnitude and DirectionTrigonometric FunctionsCoordinate SystemRadians and Degrees
Magnitude and Direction
To begin with, understanding vector components involves grasping the concepts of magnitude and direction. Magnitude refers to the size or length of the vector and is usually denoted as ||v|| or simply as a numerical value with respective unit measures. For example, if a vector has a magnitude of 4√3, it means the length of the vector is four times the square root of three.
Direction, on the other hand, indicates where the vector is pointing. This is usually expressed using an angle θ (theta), which is measured from the positive x-axis in a counter-clockwise direction within a coordinate system. To fully understand a vector's properties, you need to know both of these attributes. When given in degrees, angles often need to be converted into radians, particularly when using trigonometric functions to find the vectors' components.
For example, a vector with a magnitude ||v|| = 4√3 and an angle θ = 90° has a specific direction upwards from the positive x-axis and a fixed length in a coordinate plane.
Direction, on the other hand, indicates where the vector is pointing. This is usually expressed using an angle θ (theta), which is measured from the positive x-axis in a counter-clockwise direction within a coordinate system. To fully understand a vector's properties, you need to know both of these attributes. When given in degrees, angles often need to be converted into radians, particularly when using trigonometric functions to find the vectors' components.
For example, a vector with a magnitude ||v|| = 4√3 and an angle θ = 90° has a specific direction upwards from the positive x-axis and a fixed length in a coordinate plane.
Trigonometric Functions
Trigonometric functions are fundamental in breaking down a vector into its x and y components. These components reveal the extent of a vector's reach in each dimension of a Cartesian coordinate system. The principal functions used are cosine (cos) for the x-component and sine (sin) for the y-component.
To determine these components, you use the vector's magnitude and the direction angle. For instance, if the angle is 90 degrees, it translates to π/2 radians. The cosine of 90 degrees (or π/2 radians) is 0, resulting in a 0 x-component, while the sine of 90 degrees (or π/2 radians) is 1, which, when multiplied by the magnitude, gives the y-component.
To determine these components, you use the vector's magnitude and the direction angle. For instance, if the angle is 90 degrees, it translates to π/2 radians. The cosine of 90 degrees (or π/2 radians) is 0, resulting in a 0 x-component, while the sine of 90 degrees (or π/2 radians) is 1, which, when multiplied by the magnitude, gives the y-component.
Applying Trigonometric Functions
Given the magnitude ||v|| = 4√3 and angle θ = 90°, the components would be:- vx = ||v|| cos(θ) = 4√3 * cos(90°) = 0
- vy = ||v|| sin(θ) = 4√3 * sin(90°) = 4√3
Coordinate System
The coordinate system is essentially the stage where vectors come to life. The most commonly used system is the two-dimensional Cartesian coordinate system, which consists of the x-axis (horizontal) and y-axis (vertical). The intersection of these two axes at the point (0, 0) is known as the origin.
Vectors are depicted as arrows starting from the origin and stretching out to a point in the coordinate plane denoting their direction and magnitude. In terms of vector components, each vector can be broken down into a horizontal component along the x-axis and a vertical component along the y-axis. These components are aptly represented as vx and vy, respectively.
In the exercise, for a vector with a magnitude of 4√3 and angled at 90 degrees, the representation would be a vertical line starting at the origin and extending upward because the x-component is zero while the y-component is 4√3. This effectively positions the vector entirely on the y-axis.
Vectors are depicted as arrows starting from the origin and stretching out to a point in the coordinate plane denoting their direction and magnitude. In terms of vector components, each vector can be broken down into a horizontal component along the x-axis and a vertical component along the y-axis. These components are aptly represented as vx and vy, respectively.
In the exercise, for a vector with a magnitude of 4√3 and angled at 90 degrees, the representation would be a vertical line starting at the origin and extending upward because the x-component is zero while the y-component is 4√3. This effectively positions the vector entirely on the y-axis.
Radians and Degrees
The measurement of angles can be in either degrees or radians. These are two different units for gauging the size of an angle. One complete revolution around a circle is equal to 360 degrees or 2π radians. The conversion between degrees and radians is crucial in vector analysis and trigonometry.
To convert degrees to radians, you multiply the number of degrees by (π/180). Conversely, to convert radians to degrees, you multiply the number of radians by (180/π).
In the context of the given exercise, converting the angle of 90° to radians is necessary for using trigonometric functions, which usually operate in radian measure. Therefore, 90° converts to π/2 radians. This conversion enables us to calculate the sine and cosine correctly to determine the vector's components, hence it's a vital step in the process of depicting vectors in a coordinate system.
To convert degrees to radians, you multiply the number of degrees by (π/180). Conversely, to convert radians to degrees, you multiply the number of radians by (180/π).
In the context of the given exercise, converting the angle of 90° to radians is necessary for using trigonometric functions, which usually operate in radian measure. Therefore, 90° converts to π/2 radians. This conversion enables us to calculate the sine and cosine correctly to determine the vector's components, hence it's a vital step in the process of depicting vectors in a coordinate system.
Other exercises in this chapter
Problem 72
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