교과서 질문
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 = 2i - 8j
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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4장, 문제 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 = (7i - 3j) - (10i - 3j)
검증된 단계별 안내1
First, simplify the given vector expression by subtracting the components of the vectors: \(v = (7\mathbf{i} - 3\mathbf{j}) - (10\mathbf{i} - 3\mathbf{j})\). This means subtract the \(i\) components and the \(j\) components separately.
Calculate the resulting vector components: \(v = (7 - 10)\mathbf{i} + (-3 - (-3))\mathbf{j}\). Simplify these to find the components of \(v\).
Find the magnitude \(||v||\) of the vector using the formula \(||v|| = \sqrt{v_x^2 + v_y^2}\), where \(v_x\) and \(v_y\) are the \(i\) and \(j\) components of the vector respectively.
Determine the direction angle \(\theta\) of the vector relative to the positive \(x\)-axis using the formula \(\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)\). Make sure to consider the quadrant in which the vector lies to find the correct angle.
Convert the angle \(\theta\) from radians to degrees if necessary, and round the magnitude to the nearest hundredth and the angle to the nearest tenth of a degree as requested.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Vector Subtraction
Vector subtraction involves subtracting corresponding components of two vectors. For vectors in component form, subtract the i-components and j-components separately to find the resultant vector. This operation is essential to simplify the given vector expression before further calculations.
추천 영상:
05:29
Adding Vectors Geometrically
Magnitude of a Vector
The magnitude of a vector v = (x, y) is the length of the vector, calculated using the Pythagorean theorem as ||v|| = √(x² + y²). This scalar value represents the distance from the origin to the point (x, y) in the plane.
추천 영상:
04:44Finding Magnitude of a Vector
Direction Angle of a Vector
The direction angle θ of a vector is the angle it makes with the positive x-axis, found using θ = arctangent(y/x). It is usually measured in degrees and indicates the vector's orientation in the plane, requiring adjustment based on the vector's quadrant.
추천 영상:
05:13Finding Direction of a Vector
관련 실천
교과서 질문
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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교과서 질문
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 = (4i - 2j) - (4i - 8j)
833
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