Domanda del libro di testo
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
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 4, Problema 61
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
Guida verificata passo dopo passo1
Identify the components of the vector \( \mathbf{v} = -10\mathbf{i} + 15\mathbf{j} \). Here, the \( x \)-component is \( -10 \) and the \( y \)-component is \( 15 \).
Calculate the magnitude \( ||\mathbf{v}|| \) of the vector using the formula for the length of a vector in two dimensions:
\[ ||\mathbf{v}|| = \sqrt{(-10)^2 + 15^2} \]
Find the direction angle \( \theta \) of the vector relative to the positive \( x \)-axis using the inverse tangent function:
\[ \theta = \tan^{-1} \left( \frac{15}{-10} \right) \]
Since the \( x \)-component is negative and the \( y \)-component is positive, the vector lies in the second quadrant. Adjust the angle \( \theta \) accordingly by adding 180 degrees if necessary to get the correct direction angle measured counterclockwise from the positive \( x \)-axis.
Express the final answers: the magnitude \( ||\mathbf{v}|| \) rounded to the nearest hundredth, and the direction angle \( \theta \) rounded to the nearest tenth of a degree.
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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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Percorso guidato
04:44
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), adjusting for the correct quadrant based on the signs of a and b to ensure an accurate angle between 0° and 360°.
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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, as they represent the vector's projection on the coordinate axes.
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Percorso guidato
03:55Position Vectors & Component Form
Pratica correlata
Domanda del libro di testo
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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Domanda del libro di testo
In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 1/2, θ = 113°
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Domanda del libro di testo
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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Domanda del libro di testo
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)
486
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Domanda del libro di testo
In Exercises 53–56, let
u = -2i + 3j, v = 6i - j, w = -3i.
Find each specified vector or scalar.
4u - (2v - w)
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