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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 31
Chapter 4, Problem 31

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4j

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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., \( \mathbf{v} \cdot \mathbf{w} = 0 \).
Express the vectors \( \mathbf{v} \) and \( \mathbf{w} \) in component form. Given \( \mathbf{v} = 3\mathbf{i} \), this corresponds to \( \mathbf{v} = (3, 0) \). Similarly, \( \mathbf{w} = -4\mathbf{j} \) corresponds to \( \mathbf{w} = (0, -4) \).
Use the formula for the dot product of two vectors in component form: \[ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \] 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: \[ (3)(0) + (0)(-4) \].
Evaluate the expression to check if the dot product equals zero. If it does, then \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal; otherwise, they are not.

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Key Concepts

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

Dot Product

The dot product of two vectors is a scalar 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. It measures how much one vector extends in the direction of another.
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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 it is zero.
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Introduction to Vectors

Vector Components and Unit Vectors

Vectors can be expressed in terms of unit vectors i and j, representing the x and y directions respectively. For example, v = 3i means v has components (3, 0), and w = -4j means w has components (0, -4). Understanding this helps in calculating the dot product.
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Position Vectors & Component Form
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.

-4w

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Textbook Question

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

||-2v||

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

v = -i + 2j

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 = 95, c = 125, A = 49°

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Textbook Question

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

3w + 2v

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views