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 55
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
Guida verificata passo dopo passo1
First, recall the formula for the magnitude squared of a vector: for any vector \( \mathbf{a} \), \( ||\mathbf{a}||^2 = \mathbf{a} \cdot \mathbf{a} \), where \( \cdot \) denotes the dot product.
Express the vectors \( \mathbf{u} + \mathbf{v} \) and \( \mathbf{u} - \mathbf{v} \) explicitly by adding and subtracting their components: \( \mathbf{u} + \mathbf{v} = (-2 + 6)\mathbf{i} + (3 - 1)\mathbf{j} \) and \( \mathbf{u} - \mathbf{v} = (-2 - 6)\mathbf{i} + (3 + 1)\mathbf{j} \).
Calculate \( ||\mathbf{u} + \mathbf{v}||^2 = (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v}) \) by taking the dot product of \( \mathbf{u} + \mathbf{v} \) with itself.
Similarly, calculate \( ||\mathbf{u} - \mathbf{v}||^2 = (\mathbf{u} - \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) \) by taking the dot product of \( \mathbf{u} - \mathbf{v} \) with itself.
Finally, subtract the two results to find \( ||\mathbf{u} + \mathbf{v}||^2 - ||\mathbf{u} - \mathbf{v}||^2 \). You can also use the identity \( ||\mathbf{u} + \mathbf{v}||^2 - ||\mathbf{u} - \mathbf{v}||^2 = 4(\mathbf{u} \cdot \mathbf{v}) \) to simplify the calculation.
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Vector Addition and Subtraction
Vector addition combines corresponding components of two vectors to form a new vector, while subtraction finds the difference between components. Understanding how to add and subtract vectors component-wise is essential for calculating expressions like u + v and u - v.
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Percorso guidato
05:29
Adding Vectors Geometrically
Vector Norm (Magnitude)
The norm or magnitude of a vector is the length of the vector in space, calculated as the square root of the sum of the squares of its components. For a vector a = xi + yj, ||a|| = √(x² + y²). Squared norms (||a||²) simplify calculations by avoiding the square root.
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Percorso guidato
04:44Finding Magnitude of a Vector
Properties of Dot Product and Norms
The dot product relates to vector magnitudes and angles, and satisfies the identity ||u + v||² = ||u||² + 2(u·v) + ||v||². Using this, the expression ||u + v||² - ||u - v||² simplifies to 4(u·v), linking vector norms and dot products for efficient computation.
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Percorso guidato
05:40Introduction to Dot Product
Pratica correlata
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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
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Write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given.
||v|| = 10, θ = 330°
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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
In Exercises 53–56, let
u = -2i + 3j, v = 6i - j, w = -3i.
Find each specified vector or scalar.
4u - (2v - w)
799
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