In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4i

검증된 단계별 안내

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

Write the given vectors in component form: \( \mathbf{v} = 3\mathbf{i} = (3, 0) \) and \( \mathbf{w} = -4\mathbf{i} = (-4, 0) \).

Calculate the dot product using the formula \( \mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y \).

Substitute the components into the dot product formula: \( 3 \times (-4) + 0 \times 0 \).

Evaluate the expression to check if the dot product equals zero; if it does, the vectors are orthogonal, otherwise they are not.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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 and w, it is v · w = v₁w₁ + v₂w₂ + ... . 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, indicating no directional overlap.

Vector Components and Notation

Vectors are expressed in terms of unit vectors i, j, k representing the x, y, and z axes. Understanding how to interpret and manipulate these components is essential for calculating the dot product and analyzing vector relationships.

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.v = 3i, w = -4i