Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈-4, 4√3〉

검증된 단계별 안내

Identify the components of the vector. Here, the vector is given as \(\langle -4, 4\sqrt{3} \rangle\), where the \(x\)-component is \(-4\) and the \(y\)-component is \(4\sqrt{3}\).

Calculate the magnitude of the vector using the formula \(\text{magnitude} = \sqrt{x^2 + y^2}\). Substitute the values to get \(\sqrt{(-4)^2 + (4\sqrt{3})^2}\).

Simplify the expression inside the square root by squaring each component: \((-4)^2 = 16\) and \((4\sqrt{3})^2 = 16 \times 3 = 48\). Then add these results.

Find the direction angle \(\theta\) of the vector relative to the positive \(x\)-axis using the formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Substitute the components to get \(\theta = \tan^{-1}\left(\frac{4\sqrt{3}}{-4}\right)\).

Since the \(x\)-component is negative and the \(y\)-component is positive, the vector lies in the second quadrant. Adjust the angle accordingly by adding \(180^\circ\) if necessary, and round the angle to the nearest tenth of a degree.

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주요 개념

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

Vector Components and Magnitude

A vector in two dimensions can be represented by its components along the x and y axes. The magnitude (length) of the vector is found using the Pythagorean theorem: the square root of the sum of the squares of its components. For example, for vector 〈x, y〉, magnitude = √(x² + y²).

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). Care must be taken to consider the signs of x and y to determine the correct quadrant for the angle.

Rounding and Angle Measurement

When calculating angles, results are often rounded to a specified precision, such as the nearest tenth of a degree. Angles are typically expressed in degrees for practical applications, and rounding ensures clarity and consistency in communication.

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.〈-4, 4√3〉