Textbook Question
In Exercises 1–8, use the given vectors to find v⋅w and v⋅v.v = 3i + j, w = i + 3j
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8. Vectors
Dot Product
3:49 minutesProblem 7
Textbook Question
In Exercises 1–8, use the given vectors to find v⋅w and v⋅v.v = 5i, w = j
Verified step by step guidance1
insert step 1: Understand the dot product formula. The dot product of two vectors \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{w} = c\mathbf{i} + d\mathbf{j} \) is given by \( \mathbf{v} \cdot \mathbf{w} = ac + bd \).
insert step 2: Identify the components of the vectors \( \mathbf{v} \) and \( \mathbf{w} \). Here, \( \mathbf{v} = 5\mathbf{i} + 0\mathbf{j} \) and \( \mathbf{w} = 0\mathbf{i} + 1\mathbf{j} \).
insert step 3: Calculate \( \mathbf{v} \cdot \mathbf{w} \) using the dot product formula. Substitute \( a = 5, b = 0, c = 0, \) and \( d = 1 \) into the formula: \( \mathbf{v} \cdot \mathbf{w} = 5 \times 0 + 0 \times 1 \).
insert step 4: Calculate \( \mathbf{v} \cdot \mathbf{v} \) using the dot product formula. Substitute \( a = 5, b = 0 \) into the formula: \( \mathbf{v} \cdot \mathbf{v} = 5 \times 5 + 0 \times 0 \).
insert step 5: Review the calculations to ensure understanding of how the dot product is computed for both \( \mathbf{v} \cdot \mathbf{w} \) and \( \mathbf{v} \cdot \mathbf{v} \).
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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 is a mathematical operation that takes two vectors and returns a scalar. It is calculated by multiplying the corresponding components of the vectors and summing the results. For vectors v = ai + bj and w = ci + dj, the dot product is given by v·w = ac + bd. This operation is essential for determining the angle between vectors and understanding their geometric relationships.
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Vector Components
Vectors can be expressed in terms of their components along the coordinate axes. In this case, the vector v = 5i has a component of 5 along the x-axis and none along the y-axis, while w = j has a component of 1 along the y-axis and none along the x-axis. Understanding vector components is crucial for performing operations like the dot product, as it allows for straightforward calculations based on the individual contributions of each vector.
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Unit Vectors
Unit vectors are vectors with a magnitude of one, typically used to indicate direction. The standard unit vectors in a Cartesian coordinate system are i (along the x-axis) and j (along the y-axis). In the context of the given vectors, recognizing that v = 5i and w = j are scalar multiples of unit vectors helps simplify calculations and understand the geometric interpretation of the vectors involved in the dot product.
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