In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 3i + j, w = i + 3j

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Identify the components of the vectors \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} + 3\mathbf{j} \). Here, \( \mathbf{v} = (3, 1) \) and \( \mathbf{w} = (1, 3) \).

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: \[ \mathbf{v} \cdot \mathbf{w} = 3 \times 1 + 1 \times 3 \]

Recall that \( \mathbf{v} \cdot \mathbf{v} \) is the dot product of \( \mathbf{v} \) with itself, which gives the square of its magnitude: \[ \mathbf{v} \cdot \mathbf{v} = 3 \times 3 + 1 \times 1 \]

Evaluate the sums from steps 3 and 4 to find the values of \( \mathbf{v} \cdot \mathbf{w} \) and \( \mathbf{v} \cdot \mathbf{v} \) respectively.

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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 = (v1, v2) and w = (w1, w2), the dot product is v⋅w = v1w1 + v2w2.

Vector Components and Notation

Vectors in two dimensions are often expressed in terms of unit vectors i and j, representing the x and y directions respectively. For example, v = 3i + j means the vector has components (3, 1). Understanding this notation is essential for performing operations like the dot product.

Properties of the Dot Product

The dot product is commutative, meaning v⋅w = w⋅v, and distributive over vector addition. Also, the dot product of a vector with itself, v⋅v, gives the square of its magnitude, which is useful for finding vector lengths and angles between vectors.