Domanda del libro di testo
In Exercises 22–24, sketch each vector as a position vector and find its magnitude.
v = -3i - 4j
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 4, Problema 23
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = i + j, w = i - j
Guida verificata passo dopo passo1
Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if and only if their dot product is zero, i.e., \( \mathbf{v} \cdot \mathbf{w} = 0 \).
Express the vectors \( \mathbf{v} \) and \( \mathbf{w} \) in component form. Given \( \mathbf{v} = \mathbf{i} + \mathbf{j} \), this corresponds to \( \mathbf{v} = (1, 1) \). Similarly, \( \mathbf{w} = \mathbf{i} - \mathbf{j} \) corresponds to \( \mathbf{w} = (1, -1) \).
Calculate the dot product using the formula \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \), where \( v_1, v_2 \) are components of \( \mathbf{v} \) and \( w_1, w_2 \) are components of \( \mathbf{w} \).
Substitute the components into the dot product formula: \( (1)(1) + (1)(-1) \).
Evaluate the expression to check if the dot product equals zero. If it does, then \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal; otherwise, they are not.
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Dot Product
The dot product of two vectors is a scalar calculated by multiplying corresponding components and summing the results. For vectors v = (v1, v2) and w = (w1, w2), the dot product is v1*w1 + v2*w2. It measures how much one vector extends in the direction of another.
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Percorso guidato
05:40
Introduction to Dot Product
Orthogonality of Vectors
Two vectors are orthogonal if their dot product equals zero. This means they are perpendicular to each other in the vector space. Checking orthogonality involves computing the dot product and verifying if it is zero.
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Percorso guidato
03:48Introduction to Vectors
Vector Representation in Component Form
Vectors can be expressed as components along coordinate axes, such as v = i + j representing (1,1). This form allows easy computation of operations like the dot product by working with numerical components.
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03:55Position Vectors & Component Form
Pratica correlata
Domanda del libro di testo
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
u - v
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Domanda del libro di testo
In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.
a = 63, b = 22, c = 50
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Domanda del libro di testo
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 2i + 8j, w = 4i - j
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Domanda del libro di testo
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 10, b = 30, A = 150°
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Domanda del libro di testo
In Exercises 22–24, sketch each vector as a position vector and find its magnitude.
v = -3j
848
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