Domanda del libro di testo
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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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 4, Problema 38
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
Guida verificata passo dopo passo1
Identify the components of the vectors \( \mathbf{v} = 2\mathbf{i} + 4\mathbf{j} \) and \( \mathbf{w} = 6\mathbf{i} - 11\mathbf{j} \). Here, \( \mathbf{v} = (2, 4) \) and \( \mathbf{w} = (6, -11) \).
Calculate the dot product \( \mathbf{v} \cdot \mathbf{w} \) using the formula \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \). Substitute the components to get \( 2 \times 6 + 4 \times (-11) \).
Find the magnitudes of \( \mathbf{v} \) and \( \mathbf{w} \) using the formula \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2} \) and similarly for \( \mathbf{w} \). Calculate \( \|\mathbf{v}\| = \sqrt{2^2 + 4^2} \) and \( \|\mathbf{w}\| = \sqrt{6^2 + (-11)^2} \).
Use the dot product and magnitudes to find the cosine of the angle \( \theta \) between the vectors with the formula \( \cos(\theta) = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} \).
Finally, find the angle \( \theta \) by taking the inverse cosine (arccos) of the value found in the previous step, and convert the result to degrees if necessary, rounding to the nearest tenth of a degree.
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Dot Product of Two 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, e.g., for vectors v = (v1, v2) and w = (w1, w2), v ⋅ w = v1w1 + v2w2. This operation measures how much one vector extends in the direction of another.
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05:40
Introduction to Dot Product
Magnitude of a Vector
The magnitude (or length) of a vector is the distance from the origin to the point represented by the vector. For a vector v = (v1, v2), its magnitude is |v| = √(v1² + v2²). Magnitudes are essential for normalizing vectors and calculating angles between them.
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04:44Finding Magnitude of a Vector
Angle Between Two Vectors
The angle θ between two vectors v and w can be found using the dot product formula: v ⋅ w = |v||w|cos(θ). Rearranging gives θ = cos⁻¹((v ⋅ w) / (|v||w|)). This formula relates the dot product and magnitudes to the cosine of the angle, allowing calculation of the angle in degrees or radians.
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04:33Find the Angle Between Vectors
Pratica correlata
Domanda del libro di testo
In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. C = 102°, a = 16 meters, b = 20 meters
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Domanda del libro di testo
In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w.
v = i + 2j, w = 3i + 6j
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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 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
||w - u||
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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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