Ch. 4 - Laws of Sines and Cosines; Vectors

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 4 - Laws of Sines and Cosines; Vectors

Problem 27

Chapter 4, Problem 27

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 2i - 2j, w = -i + j

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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 the vectors in component form: \( \mathbf{v} = \langle 2, -2 \rangle \) and \( \mathbf{w} = \langle -1, 1 \rangle \).

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: \( (2)(-1) + (-2)(1) \).

Simplify the expression to check if the result equals zero. If it does, the vectors are orthogonal; if not, they are not orthogonal.

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Key Concepts

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

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 of the vectors and summing the results. For vectors v = (v1, v2) and w = (w1, w2), the dot product is v1*w1 + v2*w2.

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.

Vector Components and Notation

Vectors can be expressed in terms of unit vectors i and j, representing the x and y directions respectively. For example, v = 2i - 2j corresponds to the vector (2, -2). Understanding this notation is essential for performing operations like the dot product.

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i & j Notation

Related Practice

Textbook Question

In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.

P₁ = (-3, 0), P₂ = (-2, -2)

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Textbook Question

In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.

a = 14 meters, b = 12 meters, c = 4 meters

Textbook Question

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

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Textbook Question

In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.

w - v

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Textbook Question

In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.

a = 11 yards, b = 9 yards, c = 7 yards

Textbook Question

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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