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

검증된 단계별 안내

Identify the components of the vectors \( \mathbf{v} \) and \( \mathbf{w} \). Here, \( \mathbf{v} = 5\mathbf{i} = (5, 0) \) and \( \mathbf{w} = \mathbf{j} = (0, 1) \).

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: \[ \mathbf{v} \cdot \mathbf{w} = 5 \times 0 + 0 \times 1 \]

Calculate \( \mathbf{v} \cdot \mathbf{v} \) by multiplying the components of \( \mathbf{v} \) with themselves and adding: \[ \mathbf{v} \cdot \mathbf{v} = 5 \times 5 + 0 \times 0 \]

Simplify the expressions from steps 3 and 4 to find the dot products.

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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 and w, v⋅w = v₁w₁ + v₂w₂ + ... + vₙwₙ.

Unit Vectors i and j

In two-dimensional space, i and j are standard unit vectors along the x-axis and y-axis, respectively. Vector i = (1, 0) and vector j = (0, 1). They are orthogonal, meaning their dot product is zero.

Dot Product of a Vector with Itself

The dot product of a vector with itself, v⋅v, equals the square of its magnitude. It is calculated by summing the squares of its components, which helps find the length or magnitude of the vector.