In Exercises 1–8, use the given vectors to find v⋅w and v⋅v.v = 5i, w = j

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1. 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 \( \mathbf{v} \cdot \mathbf{w} = ac + bd \).
2. insert step 2: Identify the components of the vectors \( \mathbf{v} \) and \( \mathbf{w} \). Here, \( \mathbf{v} = 5\mathbf{i} + 0\mathbf{j} \) and \( \mathbf{w} = 0\mathbf{i} + 1\mathbf{j} \).
3. 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: \( \mathbf{v} \cdot \mathbf{w} = 5 \times 0 + 0 \times 1 \).
4. insert step 4: Calculate \( \mathbf{v} \cdot \mathbf{v} \) using the dot product formula. Substitute \( a = 5, b = 0 \) into the formula: \( \mathbf{v} \cdot \mathbf{v} = 5 \times 5 + 0 \times 0 \).
5. insert step 5: Review the calculations to ensure understanding of how the dot product is computed for both \( \mathbf{v} \cdot \mathbf{w} \) and \( \mathbf{v} \cdot \mathbf{v} \).

Key Concepts

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.

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.

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.