Domanda del libro di testo
In Exercises 39–40, find h to the nearest tenth.
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 4, Problema 40
In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.
v = 12i - 8j, w = 2i + 3j
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Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if and only if their dot product is zero, i.e., \( \mathbf{v} \cdot \mathbf{w} = 0 \).
Write down the components of the vectors: \( \mathbf{v} = 12\mathbf{i} - 8\mathbf{j} \) means \( \mathbf{v} = (12, -8) \), and \( \mathbf{w} = 2\mathbf{i} + 3\mathbf{j} \) means \( \mathbf{w} = (2, 3) \).
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 values: \( \mathbf{v} \cdot \mathbf{w} = (12)(2) + (-8)(3) \).
Evaluate the expression to check if the dot product equals zero. If it does, the vectors are orthogonal; if not, they are not orthogonal.
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Dot Product of Vectors
The dot product is an algebraic operation that takes two vectors and returns a scalar. It is 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.
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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 the result is zero.
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Percorso guidato
03:48Introduction to Vectors
Vector Components and Notation
Vectors can be expressed in terms of unit vectors i and j representing the x and y directions, respectively. Understanding how to interpret and manipulate these components is essential for performing operations like the dot product.
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Percorso guidato
06:01i & j Notation
Pratica correlata
Domanda del libro di testo
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 3i - 4j
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Domanda del libro di testo
In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
5u ⋅ (3v - 4w)
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Domanda del libro di testo
In Exercises 37–39, find the dot product v ⋅ w. Then find the angle between v and w to the nearest tenth of a degree.
v = 2i + 4j, w = 6i - 11j
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Domanda del libro di testo
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 6i
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Domanda del libro di testo
In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
projᵤ (v + w)
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