Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. B = -4.0i + 4.0j

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Step 1: Understand the vector components. The vector B is given in component form as \( B = -4.0\mathbf{i} + 4.0\mathbf{j} \), where \( \mathbf{i} \) represents the x-direction and \( \mathbf{j} \) represents the y-direction. This means the x-component of the vector is \( -4.0 \) and the y-component is \( 4.0 \).

Step 2: Draw the vector. Plot the vector on a Cartesian coordinate system. Start at the origin \( (0, 0) \), move \( -4.0 \) units along the x-axis (to the left), and then move \( 4.0 \) units along the y-axis (upward). Draw an arrow from the origin to the point \( (-4.0, 4.0) \). Label the angle \( \theta \) between the vector and the negative x-axis.

Step 3: Calculate the magnitude of the vector. Use the Pythagorean theorem: \( |B| = \sqrt{(B_x)^2 + (B_y)^2} \), where \( B_x = -4.0 \) and \( B_y = 4.0 \). Substitute these values into the formula: \( |B| = \sqrt{(-4.0)^2 + (4.0)^2} \).

Step 4: Determine the direction of the vector. The direction \( \theta \) is the angle the vector makes with the positive x-axis. Use the formula \( \theta = \arctan\left(\frac{B_y}{B_x}\right) \). Substitute \( B_x = -4.0 \) and \( B_y = 4.0 \): \( \theta = \arctan\left(\frac{4.0}{-4.0}\right) \). Note that the vector lies in the second quadrant, so adjust the angle accordingly to reflect its position.

Step 5: Express the final magnitude and direction. The magnitude is the result from Step 3, and the direction is the adjusted angle from Step 4. Ensure the direction is expressed in degrees or radians, as required, and clearly indicates the vector's orientation relative to the positive x-axis.

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주요 개념

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

Vector Representation

Vectors are quantities that have both magnitude and direction, represented in a coordinate system. In this case, the vector B = -4.0i + 4.0j can be visualized in a 2D Cartesian plane, where 'i' represents the x-axis and 'j' represents the y-axis. The components indicate that the vector points left (negative x-direction) and up (positive y-direction).

Magnitude of a Vector

The magnitude of a vector is a measure of its length and can be calculated using the Pythagorean theorem. For vector B, the magnitude is found by taking the square root of the sum of the squares of its components: |B| = √((-4.0)² + (4.0)²) = √(16 + 16) = √32, which simplifies to 4√2. This value represents how far the vector extends from the origin.

Direction of a Vector

The direction of a vector is specified by the angle it makes with a reference axis, typically the positive x-axis. This angle can be calculated using the arctangent function: θ = arctan(y/x). For vector B, the angle can be found as θ = arctan(4.0 / -4.0), which gives an angle in the second quadrant, indicating the vector's orientation in the Cartesian plane.