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)
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 35
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
||2u||
Verified step by step guidance1
Recall that the magnitude (or norm) of a vector \(\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j}\) is given by the formula: \(||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2}\).
First, find the vector \(2\mathbf{u}\) by multiplying each component of \(\mathbf{u} = 2\mathbf{i} - 5\mathbf{j}\) by 2: \(2\mathbf{u} = 2 \times 2\mathbf{i} + 2 \times (-5\mathbf{j})\).
Simplify the components of \(2\mathbf{u}\) to get the new vector in component form.
Apply the magnitude formula to the vector \(2\mathbf{u}\) by squaring each component, summing them, and then taking the square root: \(||2\mathbf{u}|| = \sqrt{(\text{component}_x)^2 + (\text{component}_y)^2}\).
This result gives the magnitude of the vector \(2\mathbf{u}\), which is the length of the vector scaled by 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Magnitude (Norm)
The magnitude or norm of a vector is the length of the vector in space, calculated using the Pythagorean theorem. For a vector v = ai + bj, its magnitude ||v|| = √(a² + b²). This represents the distance from the origin to the point (a, b) in the plane.
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Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a scalar value. If a vector u = ai + bj is multiplied by a scalar k, the resulting vector is ku = (ka)i + (kb)j. This operation scales the vector's magnitude by |k| without changing its direction if k > 0.
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Properties of Vector Norm under Scalar Multiplication
The magnitude of a scalar multiple of a vector satisfies ||ku|| = |k| * ||u||. This means when a vector is scaled by a scalar k, its length is multiplied by the absolute value of k. This property simplifies finding magnitudes of scaled vectors.
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Related Practice
Textbook Question
In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.
B = 125°, a = 8 yards, c = 5 yards
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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
806
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Textbook Question
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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Textbook Question
If u = 5i + 2j, v = i - j, and w = 3i - 7j, find u ⋅ (v + w).
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Textbook Question
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 + 3j, w = -2i + 5j
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