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.
-4w
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 4, Problema 29
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4i
Guida verificata passo dopo passo1
Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if their dot product is zero, i.e., \( \mathbf{v} \cdot \mathbf{w} = 0 \).
Write the given vectors in component form: \( \mathbf{v} = 3\mathbf{i} = (3, 0) \) and \( \mathbf{w} = -4\mathbf{i} = (-4, 0) \).
Calculate the dot product using the formula \( \mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y \).
Substitute the components into the dot product formula: \( 3 \times (-4) + 0 \times 0 \).
Evaluate the expression to check if the dot product equals zero; if it does, the vectors 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 and w, it is v · w = v₁w₁ + v₂w₂ + ... . 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, indicating no directional overlap.
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Percorso guidato
03:48Introduction to Vectors
Vector Components and Notation
Vectors are expressed in terms of unit vectors i, j, k representing the x, y, and z axes. Understanding how to interpret and manipulate these components is essential for calculating the dot product and analyzing vector relationships.
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Percorso guidato
06:01i & j Notation
Pratica correlata
Domanda del libro di testo
In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.
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Domanda del libro di testo
In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
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Domanda del libro di testo
In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.
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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 = 7, b = 28, A = 12°
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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 = 95, c = 125, A = 49°
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