Textbook Question
In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.
v = 12i - 8j, w = 2i + 3j
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 39
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 6i
Verified step by step guidance1
Identify the given vector \( \mathbf{v} = 6\mathbf{i} \). This means the vector has components \( (6, 0) \) in the 2D plane, where \( \mathbf{i} \) is the unit vector along the x-axis.
Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula for the magnitude of a vector:
\[ \\| \mathbf{v} \\| = \sqrt{(6)^2 + (0)^2} \]
Simplify the magnitude expression to find the length of \( \mathbf{v} \). This will give you a scalar value representing how long the vector is.
To find the unit vector in the same direction as \( \mathbf{v} \), divide each component of \( \mathbf{v} \) by its magnitude:
\[ \mathbf{u} = \frac{1}{\| \mathbf{v} \\|} \mathbf{v} = \left( \frac{6}{\| \mathbf{v} \\|}, \frac{0}{\| \mathbf{v} \\|} \right) \]
Express the unit vector \( \mathbf{u} \) clearly, showing that it has the same direction as \( \mathbf{v} \) but a magnitude of 1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Direction
The direction of a vector is the orientation it points in space, independent of its magnitude. Two vectors have the same direction if one is a scalar multiple of the other. Understanding direction helps in finding unit vectors that preserve this orientation.
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Finding Direction of a Vector
Magnitude (Norm) of a Vector
The magnitude of a vector is its length, calculated as the square root of the sum of the squares of its components. For a vector v = 6i, the magnitude is |v| = 6. This value is essential for normalizing the vector to find a unit vector.
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04:44Finding Magnitude of a Vector
Unit Vector
A unit vector has a magnitude of 1 and indicates direction only. To find a unit vector in the same direction as v, divide v by its magnitude. This process normalizes the vector, producing a vector with length one but the same direction.
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04:04Unit Vector in the Direction of a Given Vector
Related Practice
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 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
5u ⋅ (3v - 4w)
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 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
projᵤ (v + w)