Let A = 4i - 2j, B = -3i + 5j, and E = 2A + 3B. What are the magnitude and direction of vector E?

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Start by calculating the vector E using the given formula: \( \mathbf{E} = 2\mathbf{A} + 3\mathbf{B} \). Substitute \( \mathbf{A} = 4\mathbf{i} - 2\mathbf{j} \) and \( \mathbf{B} = -3\mathbf{i} + 5\mathbf{j} \) into the equation.

Distribute the scalar multipliers to the components of \( \mathbf{A} \) and \( \mathbf{B} \): \( 2\mathbf{A} = 2(4\mathbf{i} - 2\mathbf{j}) = 8\mathbf{i} - 4\mathbf{j} \) and \( 3\mathbf{B} = 3(-3\mathbf{i} + 5\mathbf{j}) = -9\mathbf{i} + 15\mathbf{j} \).

Add the corresponding components of \( 2\mathbf{A} \) and \( 3\mathbf{B} \) to find \( \mathbf{E} \): \( \mathbf{E} = (8\mathbf{i} - 4\mathbf{j}) + (-9\mathbf{i} + 15\mathbf{j}) = (-1\mathbf{i} + 11\mathbf{j}) \).

To find the magnitude of \( \mathbf{E} \), use the formula \( |\mathbf{E}| = \sqrt{E_x^2 + E_y^2} \), where \( E_x = -1 \) and \( E_y = 11 \). Substitute these values into the formula: \( |\mathbf{E}| = \sqrt{(-1)^2 + 11^2} \).

To find the direction of \( \mathbf{E} \), calculate the angle \( \theta \) it makes with the positive x-axis using \( \theta = \arctan\left(\frac{E_y}{E_x}\right) \). Substitute \( E_x = -1 \) and \( E_y = 11 \) into the formula: \( \theta = \arctan\left(\frac{11}{-1}\right) \). Adjust the angle based on the quadrant of \( \mathbf{E} \).

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주요 개념

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

Vector Addition

Vector addition involves combining two or more vectors to produce a resultant vector. This is done by adding their corresponding components. For example, if vector A has components (Ax, Ay) and vector B has components (Bx, By), the resultant vector R is given by R = (Ax + Bx)i + (Ay + By)j.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector represented as A = Ai + Aj, the magnitude |A| is given by |A| = √(Ax² + Ay²). This value indicates how far the vector extends in space, regardless of its direction.

Direction of a Vector

The direction of a vector indicates the orientation in which it acts and is often expressed as an angle relative to a reference axis. For a vector A = Ai + Aj, the direction can be found using the tangent function: θ = arctan(Ay/Ax). This angle helps in understanding how the vector is positioned in a coordinate system.