Let A=(3.0 m,20∘ south of east),B=(2.0 m,north)\mathbf{A} = (3.0 \, \text{m}, 20^\circ \, \text{south of east}), \quad \mathbf{B} = (2.0 \, \text{m}, \text{north}), and C=(5.0 m,70∘ south of west)\mathbf{C} = (5.0 \, \text{m}, 70^\circ \, \text{south of west}). Find the magnitude and the direction of D=A+B+C\mathbf{D} = \mathbf{A} + \mathbf{B} + \mathbf{C}.

Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

문제 22c

3장, 문제 22c

Let $A = (3.0 m, 20^{\circ} south of east),$ , and $C = (5.0 m, 70^{\circ} south of west)$ . Find the magnitude and the direction of $D = A + B + C$ .

검증된 단계별 안내

Break each vector into its components. For vector A, calculate the x-component as A_x = A * cos(20°) and the y-component as A_y = -A * sin(20°) (negative because it is south of east). For vector B, since it points directly north, its x-component is B_x = 0 and its y-component is B_y = B. For vector C, calculate the x-component as C_x = -C * cos(70°) (negative because it is west) and the y-component as C_y = -C * sin(70°) (negative because it is south).

Add the x-components of all three vectors to find the x-component of D: D_x = A_x + B_x + C_x.

Add the y-components of all three vectors to find the y-component of D: D_y = A_y + B_y + C_y.

Calculate the magnitude of vector D using the Pythagorean theorem: |D| = sqrt(D_x^2 + D_y^2).

Determine the direction of vector D by calculating the angle θ using the formula θ = arctan(D_y / D_x). Adjust the angle based on the signs of D_x and D_y to ensure it is in the correct quadrant.

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주요 개념

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

Vector Addition

Vector addition involves combining multiple vectors to find a resultant vector. This process requires breaking each vector into its components, typically along the x (horizontal) and y (vertical) axes. The components are then summed separately to obtain the total x and y components of the resultant vector.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential for resolving vectors into their components based on their angles. For a vector at an angle θ, the x-component can be found using the cosine function (A_x = A * cos(θ)), and the y-component using the sine function (A_y = A * sin(θ)). This allows for accurate calculations of vector magnitudes and directions.

Magnitude and Direction of a Vector

The magnitude of a vector represents its length or size, while the direction indicates the angle at which it acts. After calculating the resultant vector's components, the magnitude can be found using the Pythagorean theorem (|D| = √(D_x² + D_y²)), and the direction can be determined using the arctangent function (θ = arctan(D_y / D_x)), which gives the angle relative to a reference axis.