In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 2i - 2j, w = -i + j

검증된 단계별 안내

Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if and only if their dot product is zero, i.e., \( \mathbf{v} \cdot \mathbf{w} = 0 \).

Write the vectors in component form: \( \mathbf{v} = \langle 2, -2 \rangle \) and \( \mathbf{w} = \langle -1, 1 \rangle \).

Calculate the dot product using the formula \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \), where \( v_1, v_2 \) are components of \( \mathbf{v} \) and \( w_1, w_2 \) are components of \( \mathbf{w} \).

Substitute the components into the dot product formula: \( (2)(-1) + (-2)(1) \).

Simplify the expression to check if the result equals zero. If it does, the vectors are orthogonal; if not, they are not orthogonal.

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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 v1*w1 + v2*w2.

Orthogonality of Vectors

Two vectors are orthogonal if their dot product equals zero. This means they are perpendicular to each other in the vector space. Checking orthogonality involves computing the dot product and verifying if the result is zero.

Vector Components and Notation

Vectors can be expressed in terms of unit vectors i and j, representing the x and y directions respectively. For example, v = 2i - 2j corresponds to the vector (2, -2). Understanding this notation is essential for performing operations like the dot product.