Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.
〈8√2, -8√2〉

검증된 단계별 안내

Identify the components of the vector. Here, the vector is given as \(\langle 8\sqrt{2}, -8\sqrt{2} \rangle\), where the x-component is \(8\sqrt{2}\) and the y-component is \(-8\sqrt{2}\).

Calculate the magnitude of the vector using the formula \(\text{magnitude} = \sqrt{x^2 + y^2}\). Substitute the components: \(\sqrt{(8\sqrt{2})^2 + (-8\sqrt{2})^2}\).

Simplify the expression inside the square root by squaring each component and adding them together.

Find the direction angle \(\theta\) using the formula \(\theta = \tan^{-1}\left( \frac{y}{x} \right)\). Substitute the components: \(\theta = \tan^{-1}\left( \frac{-8\sqrt{2}}{8\sqrt{2}} \right)\).

Determine the correct quadrant for the angle based on the signs of the x and y components, then adjust the angle accordingly to find the direction angle measured counterclockwise from the positive x-axis. Round the angle to the nearest tenth of a degree.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Vector Magnitude

The magnitude of a vector represents its length and is calculated using the Pythagorean theorem. For a vector with components (x, y), the magnitude is √(x² + y²). This gives a non-negative scalar value indicating the vector's size regardless of direction.

Direction Angle of a Vector

The direction angle of a vector is the angle it makes with the positive x-axis, measured counterclockwise. It can be found using the inverse tangent function: θ = arctan(y/x). Adjustments may be needed based on the vector's quadrant to get the correct angle.

Handling Vectors with Radical Components

Vectors with components involving radicals, like √2, require careful arithmetic when calculating magnitude and direction. Simplifying expressions and using exact values before rounding helps maintain accuracy in the final results.

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.〈8√2, -8√2〉