Textbook Question
Use the figure to find each vector: u - v. Use vector notation as in Example 4.
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8. Vectors
Geometric Vectors
Problem 7.31c
Textbook Question
Use the figure to find each vector: - u. Use vector notation as in Example 4.
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Verified step by step guidance1
Step 1: Identify the vector \( \mathbf{u} \) from the given figure. Note its direction and magnitude.
Step 2: Determine the components of vector \( \mathbf{u} \). If \( \mathbf{u} \) is represented as \( \langle a, b \rangle \), identify \( a \) and \( b \) from the figure.
Step 3: To find \( -\mathbf{u} \), negate each component of \( \mathbf{u} \). This means if \( \mathbf{u} = \langle a, b \rangle \), then \( -\mathbf{u} = \langle -a, -b \rangle \).
Step 4: Write the vector \( -\mathbf{u} \) in vector notation as \( \langle -a, -b \rangle \).
Step 5: Verify the direction of \( -\mathbf{u} \) is opposite to that of \( \mathbf{u} \) by checking the signs of the components.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Notation
Vector notation is a mathematical representation of vectors, typically denoted as an ordered pair or triplet of numbers that indicate its components along the coordinate axes. For example, a vector u in two dimensions can be represented as u = <x, y>, where x and y are the horizontal and vertical components, respectively. Understanding this notation is essential for accurately describing and manipulating vectors in trigonometry.
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Vector Addition
Vector addition is the process of combining two or more vectors to produce a resultant vector. This is done by adding the corresponding components of the vectors. For instance, if vector u = <x1, y1> and vector v = <x2, y2>, their sum is u + v = <x1 + x2, y1 + y2>. Mastery of vector addition is crucial for solving problems involving multiple vectors and understanding their interactions.
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Magnitude of a Vector
The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector u = <x, y>, the magnitude is given by ||u|| = √(x² + y²). This concept is important for understanding the size of a vector in relation to its direction, which is fundamental in trigonometry and physics when analyzing forces and motion.
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