In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = i + j, 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., \( \mathbf{v} \cdot \mathbf{w} = 0 \).

Express the vectors \( \mathbf{v} \) and \( \mathbf{w} \) in component form. Given \( \mathbf{v} = \mathbf{i} + \mathbf{j} \), this corresponds to \( \mathbf{v} = (1, 1) \). Similarly, \( \mathbf{w} = \mathbf{i} - \mathbf{j} \) corresponds to \( \mathbf{w} = (1, -1) \).

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: \( (1)(1) + (1)(-1) \).

Evaluate the expression to check if the dot product equals zero. If it does, then \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal; otherwise, they are not.

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주요 개념

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

Dot Product

The dot product of two vectors is a scalar calculated by multiplying corresponding components and summing the results. For vectors v = (v1, v2) and w = (w1, w2), the dot product is v1*w1 + v2*w2. It measures how much one vector extends in the direction of another.

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 it is zero.

Vector Representation in Component Form

Vectors can be expressed as components along coordinate axes, such as v = i + j representing (1,1). This form allows easy computation of operations like the dot product by working with numerical components.

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