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 33

Chapter 4, Problem 33

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 = 3i - 2j, w = i - j

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Identify the vectors \( \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \) and \( \mathbf{w} = \mathbf{i} - \mathbf{j} \).

Recall the formula for the projection of \( \mathbf{v} \) onto \( \mathbf{w} \): \[ \text{proj}_{\mathbf{w}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \right) \mathbf{w} \] where \( \cdot \) denotes the dot product.

Calculate the dot product \( \mathbf{v} \cdot \mathbf{w} \) by multiplying corresponding components and summing: \[ \mathbf{v} \cdot \mathbf{w} = (3)(1) + (-2)(-1) \]

Calculate the dot product \( \mathbf{w} \cdot \mathbf{w} \) similarly: \[ \mathbf{w} \cdot \mathbf{w} = (1)(1) + (-1)(-1) \]

Use the values from the dot products to find \( \text{proj}_{\mathbf{w}} \mathbf{v} \) by multiplying the scalar \( \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \) by the vector \( \mathbf{w} \). Then, find \( \mathbf{v}_1 = \text{proj}_{\mathbf{w}} \mathbf{v} \) (parallel component) and \( \mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1 \) (orthogonal component).

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Key Concepts

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

Vector Projection

Vector projection of v onto w, denoted proj_w v, is the component of v that points in the direction of w. It is calculated using the formula proj_w v = (v · w / w · w) w, where '·' denotes the dot product. This concept helps in finding how much of one vector lies along another.

Dot Product of Vectors

The dot product is an algebraic operation that takes two vectors and returns a scalar. It is computed as v · w = v₁w₁ + v₂w₂ for 2D vectors. The dot product is essential for finding projections and determining angles between vectors.

Vector Decomposition into Parallel and Orthogonal Components

Any vector v can be decomposed into two components: v₁ parallel to w and v₂ orthogonal to w. Here, v₁ = proj_w v, and v₂ = v - v₁. This decomposition is useful in many applications, such as resolving forces or simplifying vector problems.

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Related Practice

Textbook Question

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

3v - 4w

Textbook Question

In Exercises 35–36, the three given points are the vertices of a triangle. Solve each triangle, rounding lengths of sides to the nearest tenth and angle measures to the nearest degree.

A(0, 0), B(-3, 4), C(3, -1)

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 = 1.4, b = 2.9, A = 142°

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Textbook Question

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.

A = 22°, b = 20 feet, c = 50 feet

Textbook Question

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

||2u||

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Textbook Question

In Exercises 31–32, find the unit vector that has the same direction as the vector v.

v = -i + 2j