Problem 74
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}\|=2} & {\mathbf{v} \text { in the direction } \mathbf{i}+3 \mathbf{j}}\end{array}
Step-by-Step Solution
Verified Answer
The component form of the given vector is \(v = 2\cos(71.57)\mathbf{i} + 2\sin(71.57)\mathbf{j}\)
1Step 1: Understanding the Problem
We are given that the vector \(v\) has a magnitude of 2 and is in the direction \(\mathbf{i} + 3\mathbf{j}\) which represents a vector in the xy-plane pointing in the positive x-direction and 3 times more in the positive y-direction. Our task is to find the component form of the given vector.
2Step 2: Determining the Angle
In order to determine the angle that \(v\) forms with the positive x-axis, we can use the following formula: \(\theta = \arctan\left(\frac{rise}{run}\right)\) = \(\arctan\left(\frac{3}{1}\right)\) where the rise corresponds to the j-component and run corresponds to the i-component. Using a calculator, we find that \(\arctan(\frac{3}{1}) \approx 71.57^\circ.\) Hence, the angle is approximately 71.57 degrees.
3Step 3: Computing the Component form of the Vector
The x-component and y-component of the vector can be found using trigonometric relationships. The x-component (or i-component) can be found by multplying the magnitude of the vector by the cosine of the angle: x = magnitude * cos(θ) = 2 * cos (71.57) The y-component (or j-component) can be found by multiplying the magnitude of the vector by the sine of the angle: y = magnitude * sin(θ) = 2 * sin (71.57). Therefore, the component form of the vector can be expressed as \(v = 2\cos(71.57)\mathbf{i} + 2\sin(71.57)\mathbf{j}\)
Key Concepts
Magnitude of VectorAngle with X-axisTrigonometric Functions
Magnitude of Vector
Understanding the magnitude of a vector is essential in the world of physics and mathematics. It is a measure of how long the vector is, or in other words, the distance from the vector's starting point to its endpoint, regardless of direction.
A vector can be represented in two forms: graphically, as an arrow in a space, or algebraically, as an ordered pair (or triplet, quadruplet, etc., depending on the number of dimensions) of numbers. These numbers correspond to the vector’s components along the respective axes of the system.
The magnitude, often symbolized by vertical bars like \( \|\vec{v}\| \), can be found using the Pythagorean theorem if the components are known: \( \|\vec{v}\| = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the components along the horizontal (x-axis) and vertical (y-axis) directions, respectively. For three-dimensional vectors, the formula extends to include the z-component: \( \|\vec{v}\| = \sqrt{x^2 + y^2 + z^2} \).
The magnitude provides a means to compare vectors objectively, as it does not depend on the vector's direction—only its length.
A vector can be represented in two forms: graphically, as an arrow in a space, or algebraically, as an ordered pair (or triplet, quadruplet, etc., depending on the number of dimensions) of numbers. These numbers correspond to the vector’s components along the respective axes of the system.
The magnitude, often symbolized by vertical bars like \( \|\vec{v}\| \), can be found using the Pythagorean theorem if the components are known: \( \|\vec{v}\| = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the components along the horizontal (x-axis) and vertical (y-axis) directions, respectively. For three-dimensional vectors, the formula extends to include the z-component: \( \|\vec{v}\| = \sqrt{x^2 + y^2 + z^2} \).
The magnitude provides a means to compare vectors objectively, as it does not depend on the vector's direction—only its length.
Angle with X-axis
The angle a vector makes with the x-axis is a critical piece of information when trying to understand its direction relative to a coordinate system. This angle is measured from the positive x-axis to the vector and is usually represented by the Greek letter \( \theta \) (theta).
To find this angle, one can use inverse trigonometric functions, specifically the arctangent function if the vector's components are known: \( \theta = \arctan\left(\frac{rise}{run}\right) \), where 'rise' denotes movement along the y-axis and 'run' corresponds to movement along the x-axis. This ratio gives the tangent of the angle \( \theta \) in a right-angled triangle.
In our exercise, by treating the given direction \( \mathbf{i}+3\mathbf{j} \) as a vector's components, we can interpret 'run' as the coefficient of \( \mathbf{i} \) (which is 1) and 'rise' as the coefficient of \( \mathbf{j} \) (which is 3). Plugging these into the arctangent function provides the angle in degrees or radians—whichever unit is preferred for the study at hand. Always remember that the angle measured should be between 0 and 180 degrees if working in the xy-plane.
To find this angle, one can use inverse trigonometric functions, specifically the arctangent function if the vector's components are known: \( \theta = \arctan\left(\frac{rise}{run}\right) \), where 'rise' denotes movement along the y-axis and 'run' corresponds to movement along the x-axis. This ratio gives the tangent of the angle \( \theta \) in a right-angled triangle.
In our exercise, by treating the given direction \( \mathbf{i}+3\mathbf{j} \) as a vector's components, we can interpret 'run' as the coefficient of \( \mathbf{i} \) (which is 1) and 'rise' as the coefficient of \( \mathbf{j} \) (which is 3). Plugging these into the arctangent function provides the angle in degrees or radians—whichever unit is preferred for the study at hand. Always remember that the angle measured should be between 0 and 180 degrees if working in the xy-plane.
Trigonometric Functions
Trigonometric functions are fundamental tools in mathematics, especially when it comes to dealing with angles and circles. They link the angles of a triangle to the lengths of its sides and are crucial in many areas of study, including physics, engineering, and even computer graphics.
The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Each serves a different purpose when analyzing a vector: sine relates to the y-component, cosine to the x-component, and tangent is the ratio of the sine to the cosine.
In our context, when calculating the component form of a vector given its magnitude and the angle it makes with the x-axis, we multiply the magnitude by the cosine of the angle to get the x-component, and by the sine of the angle to find the y-component:
The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Each serves a different purpose when analyzing a vector: sine relates to the y-component, cosine to the x-component, and tangent is the ratio of the sine to the cosine.
In our context, when calculating the component form of a vector given its magnitude and the angle it makes with the x-axis, we multiply the magnitude by the cosine of the angle to get the x-component, and by the sine of the angle to find the y-component:
- X-component: \( x = \text{magnitude} \times \cos(\theta) \)
- Y-component: \( y = \text{magnitude} \times \sin(\theta) \)
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