In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = -6i - 5j, w = -10i - 8j

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Identify the components of the vectors \( \mathbf{v} = -6\mathbf{i} - 5\mathbf{j} \) and \( \mathbf{w} = -10\mathbf{i} - 8\mathbf{j} \). Here, \( \mathbf{v} = (-6, -5) \) and \( \mathbf{w} = (-10, -8) \).

Recall the formula for the dot product of two vectors \( \mathbf{v} = (v_1, v_2) \) and \( \mathbf{w} = (w_1, w_2) \): \[ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \]

Calculate \( \mathbf{v} \cdot \mathbf{w} \) by multiplying the corresponding components and adding the results: \[ (-6)(-10) + (-5)(-8) \]

Recall that \( \mathbf{v} \cdot \mathbf{v} \) is the dot product of \( \mathbf{v} \) with itself, which gives the square of its magnitude. Use the formula: \[ \mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2 \]

Calculate \( \mathbf{v} \cdot \mathbf{v} \) by squaring each component of \( \mathbf{v} \) and adding them: \[ (-6)^2 + (-5)^2 \]

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주요 개념

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Dot Product of Vectors

The dot product is an algebraic operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components and summing the results, e.g., for vectors v = ai + bj and w = ci + dj, v⋅w = ac + bd. This operation measures the extent to which two vectors point in the same direction.

Vector Components and Notation

Vectors in two dimensions are expressed in terms of unit vectors i and j, representing the x and y directions respectively. Each vector is written as v = ai + bj, where a and b are scalar components along the x and y axes. Understanding this notation is essential for performing operations like the dot product.

Self Dot Product and Vector Magnitude

The dot product of a vector with itself, v⋅v, equals the sum of the squares of its components, which corresponds to the square of its magnitude (length). For v = ai + bj, v⋅v = a² + b². This concept is useful for finding the length of a vector or comparing vector sizes.