In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v. v = -10i + 15j

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Identify the components of the vector \( \mathbf{v} = -10\mathbf{i} + 15\mathbf{j} \). Here, the \( x \)-component is \( -10 \) and the \( y \)-component is \( 15 \).

Calculate the magnitude \( ||\mathbf{v}|| \) of the vector using the formula for the length of a vector in two dimensions: \[ ||\mathbf{v}|| = \sqrt{(-10)^2 + 15^2} \]

Find the direction angle \( \theta \) of the vector relative to the positive \( x \)-axis using the inverse tangent function: \[ \theta = \tan^{-1} \left( \frac{15}{-10} \right) \]

Since the \( x \)-component is negative and the \( y \)-component is positive, the vector lies in the second quadrant. Adjust the angle \( \theta \) accordingly by adding 180 degrees if necessary to get the correct direction angle measured counterclockwise from the positive \( x \)-axis.

Express the final answers: the magnitude \( ||\mathbf{v}|| \) rounded to the nearest hundredth, and the direction angle \( \theta \) rounded 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 Magnitude

The magnitude of a vector represents its length and is calculated using the Pythagorean theorem. For a vector v = ai + bj, the magnitude ||v|| is √(a² + b²). This gives a non-negative scalar value indicating the vector's size regardless of direction.

Direction Angle of a Vector

The direction angle θ of a vector in the plane is the angle it makes with the positive x-axis, measured counterclockwise. It can be found using θ = arctan(b/a), adjusting for the correct quadrant based on the signs of a and b to ensure an accurate angle between 0° and 360°.

Component Form of Vectors

Vectors in two dimensions are often expressed in component form as v = ai + bj, where a and b are the horizontal and vertical components, respectively. Understanding these components is essential for calculating magnitude and direction, as they represent the vector's projection on the coordinate axes.