Textbook Question
In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.P₁ = (2, -1), P₂ = (5, -3)
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8. Vectors
Geometric Vectors
1:40 minutesProblem 29
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
-4w
Verified step by step guidance1
Identify the given vector \( w = -\mathbf{i} - 6\mathbf{j} \), which can be written in component form as \( w = (-1, -6) \).
Understand that multiplying a vector by a scalar means multiplying each component of the vector by that scalar.
Set up the scalar multiplication: \( -4w = -4 \times (-1, -6) \).
Multiply each component of \( w \) by \( -4 \): \( -4 \times (-1) \) for the \( \mathbf{i} \) component and \( -4 \times (-6) \) for the \( \mathbf{j} \) component.
Write the resulting vector after multiplication as \( -4w = (4, 24) \), which corresponds to \( 4\mathbf{i} + 24\mathbf{j} \).
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation in Component Form
Vectors in two dimensions can be expressed as a combination of unit vectors i and j, representing the x and y components respectively. For example, u = 2i - 5j means the vector has an x-component of 2 and a y-component of -5. Understanding this form allows for straightforward vector operations like addition, subtraction, and scalar multiplication.
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Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). This operation changes the magnitude of the vector without altering its direction if the scalar is positive, or reverses the direction if the scalar is negative. For instance, multiplying vector w by -4 scales and reverses w.
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Vector Notation and Operations
Vectors are often denoted using unit vectors i and j for clarity in two dimensions. Operations like addition, subtraction, and scalar multiplication are performed component-wise. Recognizing how to manipulate vectors in this notation is essential for solving problems involving vector quantities.
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