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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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