In Exercises 9–16, let u = 2i - j, v = 3i + j, and w = i + 4j. Find each specified scalar. u ⋅ (v + w)

검증된 단계별 안내

First, find the vector sum \( v + w \). To do this, add the corresponding components of vectors \( v \) and \( w \).

Calculate \( v + w = (3i + j) + (i + 4j) \).

Combine the \( i \) components: \( 3i + i = 4i \).

Combine the \( j \) components: \( j + 4j = 5j \).

Now, compute the dot product \( u \cdot (v + w) \) by multiplying the corresponding components of \( u = 2i - j \) and \( v + w = 4i + 5j \), and then summing the results: \( (2 \times 4) + ((-1) \times 5) \).

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주요 개념

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

Vector Addition

Vector addition involves combining two or more vectors to form a resultant vector. In this case, the vectors v and w are added together by adding their corresponding components. For example, if v = 3i + j and w = i + 4j, their sum is (3i + i) + (j + 4j) = 4i + 5j.

Dot Product

The dot product, or scalar product, of two vectors is a way to multiply them to obtain a scalar value. It is calculated by multiplying the corresponding components of the vectors and summing the results. For vectors u = 2i - j and v = 4i + 5j, the dot product is computed as (2 * 4) + (-1 * 5) = 8 - 5 = 3.

Scalar Result

A scalar result is a single numerical value obtained from operations involving vectors, such as the dot product. In the context of the given problem, the scalar result represents the magnitude of the projection of one vector onto another, providing insight into their directional relationship. This value is crucial for applications in physics and engineering.