In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i - 4j, w = -2i - j

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Identify the components of the vectors \( \mathbf{v} = 5\mathbf{i} - 4\mathbf{j} \) and \( \mathbf{w} = -2\mathbf{i} - \mathbf{j} \). Here, \( \mathbf{v} = (5, -4) \) and \( \mathbf{w} = (-2, -1) \).

Recall the formula for the dot product of two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \): \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \]

Calculate \( \mathbf{v} \cdot \mathbf{w} \) by multiplying corresponding components and adding the results: \[ \mathbf{v} \cdot \mathbf{w} = (5)(-2) + (-4)(-1) \]

Recall that \( \mathbf{v} \cdot \mathbf{v} \) is the dot product of \( \mathbf{v} \) with itself, which gives the square of its magnitude: \[ \mathbf{v} \cdot \mathbf{v} = 5^2 + (-4)^2 \]

Perform the arithmetic operations in the expressions from steps 3 and 4 to find the values of \( \mathbf{v} \cdot \mathbf{w} \) and \( \mathbf{v} \cdot \mathbf{v} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 of the vectors and summing the results. For vectors v = ai + bj and w = ci + dj, the dot product is v·w = ac + bd.

Vector Components

Vectors in two dimensions can be expressed in terms of their components along the i (x-axis) and j (y-axis) unit vectors. Understanding how to identify and use these components is essential for operations like addition, subtraction, and dot product.

Dot Product of a Vector with Itself

The dot product of a vector with itself, v·v, gives the square of its magnitude. It is calculated by summing the squares of its components, i.e., v·v = a² + b² for v = ai + bj. This is useful for finding the length or magnitude of the vector.