For each pair of vectors u and v with angle θ between them, sketch the resultant.

|u| = 12, |v| = 20, θ = 27°

검증된 단계별 안내

Identify the given information: the magnitudes of vectors \( \mathbf{u} \) and \( \mathbf{v} \) are \( |\mathbf{u}| = 12 \) and \( |\mathbf{v}| = 20 \), and the angle between them is \( \theta = 27^\circ \).

Recall that the resultant vector \( \mathbf{R} = \mathbf{u} + \mathbf{v} \) can be found using the Law of Cosines for vectors, where the magnitude of \( \mathbf{R} \) is given by: \[ |\mathbf{R}| = \sqrt{|\mathbf{u}|^2 + |\mathbf{v}|^2 + 2 |\mathbf{u}| |\mathbf{v}| \cos(\theta)} \]

To sketch the resultant, start by drawing vector \( \mathbf{u} \) with length proportional to 12 units in any direction you choose.

Next, from the tip of \( \mathbf{u} \), draw vector \( \mathbf{v} \) at an angle of \( 27^\circ \) relative to \( \mathbf{u} \), with length proportional to 20 units.

The resultant vector \( \mathbf{R} \) is then drawn from the tail of \( \mathbf{u} \) (the starting point) to the tip of \( \mathbf{v} \) (the ending point). This vector represents the sum \( \mathbf{u} + \mathbf{v} \).

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주요 개념

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

Vector Addition

Vector addition involves combining two vectors to find their resultant by placing them head-to-tail and drawing the vector from the start of the first to the end of the second. This graphical method helps visualize the resultant vector's magnitude and direction.

Law of Cosines for Vectors

The Law of Cosines relates the magnitudes of two vectors and the angle between them to find the magnitude of their resultant: |R| = √(|u|² + |v|² + 2|u||v|cosθ). This formula is essential for calculating the exact length of the resultant vector.

Angle Between Vectors

The angle θ between two vectors determines how they combine. It affects both the magnitude and direction of the resultant vector, influencing whether the vectors reinforce or partially cancel each other.

For each pair of vectors u and v with angle θ between them, sketch the resultant.|u| = 12, |v| = 20, θ = 27°