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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 4 - Laws of Sines and Cosines; VectorsProblem 39
Chapter 4, Problem 39

In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
5u ⋅ (3v - 4w)

Verified step by step guidance
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First, express the vectors u, v, and w in component form: \( u = \langle -1, 1 \rangle \), \( v = \langle 3, -2 \rangle \), and \( w = \langle 0, -5 \rangle \).
Calculate the vector inside the parentheses: \( 3v - 4w \). Multiply each vector by the scalar and then subtract: \( 3v = 3 \times \langle 3, -2 \rangle = \langle 9, -6 \rangle \) and \( 4w = 4 \times \langle 0, -5 \rangle = \langle 0, -20 \rangle \). Then, \( 3v - 4w = \langle 9, -6 \rangle - \langle 0, -20 \rangle \).
Perform the subtraction component-wise: \( \langle 9 - 0, -6 - (-20) \rangle = \langle 9, 14 \rangle \).
Multiply the vector u by the scalar 5: \( 5u = 5 \times \langle -1, 1 \rangle = \langle -5, 5 \rangle \).
Finally, compute the dot product of \( 5u \) and \( 3v - 4w \): \( \langle -5, 5 \rangle \cdot \langle 9, 14 \rangle = (-5)(9) + (5)(14) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). This operation changes the magnitude of the vector without altering its direction unless the scalar is negative, which reverses the direction.
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Multiplying Vectors By Scalars

Vector Addition and Subtraction

Vector addition and subtraction are performed component-wise. To add or subtract vectors, add or subtract their corresponding components, resulting in a new vector that combines the effects of the original vectors.
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Adding Vectors Geometrically

Dot Product of Vectors

The dot product is a scalar obtained by multiplying corresponding components of two vectors and summing the results. It measures the extent to which two vectors point in the same direction and is calculated as u ⋅ v = u₁v₁ + u₂v₂ for 2D vectors.
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Related Practice
Textbook Question

In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.

v = 12i - 8j, w = 2i + 3j

Textbook Question

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

Textbook Question

In Exercises 39–40, find h to the nearest tenth.

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Textbook Question

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

Textbook Question

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.

||w - u||

Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = 6i