Textbook Question
In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v. v = -10i + 15j
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 62
Find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v.
v = 2i - 8j
Verified step by step guidance1
Identify the components of the vector \( \mathbf{v} = 2\mathbf{i} - 8\mathbf{j} \). Here, the x-component \( v_x = 2 \) and the y-component \( v_y = -8 \).
Calculate the magnitude \( ||\mathbf{v}|| \) using the formula for the length of a vector in two dimensions:
\[ ||\mathbf{v}|| = \sqrt{v_x^2 + v_y^2} \]
Substitute the values of \( v_x \) and \( v_y \) into this formula.
Find the direction angle \( \theta \) of the vector relative to the positive x-axis using the inverse tangent function:
\[ \theta = \tan^{-1} \left( \frac{v_y}{v_x} \right) \]
Substitute the values of \( v_y \) and \( v_x \) into this formula.
Since the vector has a negative y-component and a positive x-component, determine the correct quadrant for \( \theta \) and adjust the angle accordingly to get the direction angle measured counterclockwise from the positive x-axis.
Express the magnitude rounded to the nearest hundredth and the direction angle \( \theta \) rounded to the nearest tenth of a degree.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Magnitude
The magnitude of a vector represents its length and is calculated using the Pythagorean theorem. For a vector v = ai + bj, the magnitude ||v|| is √(a² + b²). This gives a non-negative scalar value indicating the vector's size regardless of direction.
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Finding Magnitude of a Vector
Direction Angle of a Vector
The direction angle θ of a vector in the plane is the angle it makes with the positive x-axis, measured counterclockwise. It can be found using θ = arctan(b/a), where a and b are the vector's components. Adjustments may be needed based on the quadrant.
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05:13Finding Direction of a Vector
Component Form of Vectors
Vectors in two dimensions are often expressed in component form as v = ai + bj, where a and b are the horizontal and vertical components, respectively. Understanding these components is essential for calculating magnitude and direction.
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03:55Position Vectors & Component Form
Related Practice
Textbook Question
In Exercises 53–56, let u = -2i + 3j, v = 6i - j, w = -3i. Find each specified vector or scalar. ||u + v||² - ||u - v||²
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Textbook Question
In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v. v = (4i - 2j) - (4i - 8j)
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Textbook Question
Find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v.
v = (7i - 3j) - (10i - 3j)
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Textbook Question
In Exercises 53–56, letu = -2i + 3j, v = 6i - j, w = -3i.Find each specified vector or scalar.4u - (2v - w)