In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.
v = 12i - 8j, w = 2i + 3j

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 down the components of the vectors: \( \mathbf{v} = 12\mathbf{i} - 8\mathbf{j} \) means \( \mathbf{v} = (12, -8) \), and \( \mathbf{w} = 2\mathbf{i} + 3\mathbf{j} \) means \( \mathbf{w} = (2, 3) \).

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 values: \( \mathbf{v} \cdot \mathbf{w} = (12)(2) + (-8)(3) \).

Evaluate the expression to check if the dot product 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 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. Understanding how to interpret and manipulate these components is essential for performing operations like the dot product.

In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.v = 12i - 8j, w = 2i + 3j