If u = 5i + 2j, v = i - j, and w = 3i - 7j, find u ⋅ (v + w).

검증된 단계별 안내

First, understand that the problem requires finding the dot product of vector \( u \) with the sum of vectors \( v \) and \( w \). The dot product is defined as \( \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y \) for 2D vectors.

Calculate the vector sum \( \mathbf{v} + \mathbf{w} \) by adding their corresponding components: \( (v_x + w_x) \) for the i-component and \( (v_y + w_y) \) for the j-component.

Write down the resulting vector from step 2 explicitly as \( \mathbf{v} + \mathbf{w} = (v_x + w_x)\mathbf{i} + (v_y + w_y)\mathbf{j} \).

Now, compute the dot product \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \) by multiplying the i-components of \( \mathbf{u} \) and \( \mathbf{v} + \mathbf{w} \), and the j-components of \( \mathbf{u} \) and \( \mathbf{v} + \mathbf{w} \), then summing these products: \( u_x (v_x + w_x) + u_y (v_y + w_y) \).

Express the final dot product as a sum of products of components without calculating the numerical value, which completes the setup for the solution.

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주요 개념

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

Vector Addition

Vector addition involves combining two vectors by adding their corresponding components. For example, if v = i - j and w = 3i - 7j, then v + w = (1+3)i + (-1-7)j = 4i - 8j. This operation is essential before performing the dot product in the given problem.

Dot Product of Vectors

The dot product of two vectors u and v is a scalar calculated by multiplying their corresponding components and summing the results. For vectors u = a1i + b1j and v = a2i + b2j, u ⋅ v = a1a2 + b1b2. This operation measures the extent to which two vectors point in the same direction.

Component Form of Vectors

Vectors can be expressed in component form using unit vectors i and j along the x and y axes, respectively. This form allows easy manipulation of vectors through addition, subtraction, and dot product by working directly with their components.