Let E = 2i + 3j and F = 2i - 2j. Find the magnitude of -E - 2F.

검증된 단계별 안내

Step 1: Write down the given vectors. E = 2i + 3j and F = 2i - 2j.

Step 2: Compute the expression -E. To negate a vector, multiply each component by -1. This gives -E = -2i - 3j.

Step 3: Compute the expression -2F. Multiply each component of F by -2. This gives -2F = -4i + 4j.

Step 4: Add the vectors -E and -2F. Combine the i-components and j-components separately: (-2i - 3j) + (-4i + 4j) = (-2 - 4)i + (-3 + 4)j = -6i + 1j.

Step 5: Find the magnitude of the resulting vector -6i + 1j. Use the formula for the magnitude of a vector: |V| = sqrt((x-component)^2 + (y-component)^2). Substitute the components: |V| = sqrt((-6)^2 + (1)^2).

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주요 개념

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

Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors by adding or subtracting their corresponding components. For example, if vector E has components (2, 3) and vector F has components (2, -2), then -E would be (-2, -3) and -2F would be (-4, 4). The resultant vector is found by summing these component-wise.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y²) for a 2D vector with components (x, y). This value represents the distance from the origin to the point defined by the vector in the Cartesian plane. For example, the magnitude of vector (a, b) is √(a² + b²).

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For instance, multiplying vector F by -2 scales its components by -2, effectively reversing its direction and doubling its length. This operation is crucial for manipulating vectors in various physics problems.