Let A = 4i - 2j, B = -3i + 5j, and F = A - 4B. Write vector F in component form.

검증된 단계별 안내

Step 1: Understand the problem. You are given two vectors, A and B, in component form, and you need to calculate a new vector F using the formula F = A - 4B. This involves vector subtraction and scalar multiplication.

Step 2: Write down the components of vector A and vector B. Vector A = 4i - 2j can be expressed as (4, -2), and vector B = -3i + 5j can be expressed as (-3, 5).

Step 3: Perform scalar multiplication on vector B. Multiply each component of vector B by the scalar 4: 4 * (-3) = -12 for the i-component, and 4 * 5 = 20 for the j-component. This gives 4B = (-12, 20).

Step 4: Subtract 4B from A to find vector F. Subtract the corresponding components: For the i-component, 4 - (-12) = 4 + 12 = 16. For the j-component, -2 - 20 = -22. This gives F = (16, -22).

Step 5: Write vector F in component form. The final result is F = 16i - 22j, which is expressed as (16, -22) in component form.

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주요 개념

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Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors by adding or subtracting their corresponding components. For example, if vector A has components (Ax, Ay) and vector B has components (Bx, By), the resultant vector C = A + B will have components (Ax + Bx, Ay + By). This principle is essential for calculating vector F in the given problem.

Component Form of a Vector

The component form of a vector expresses it in terms of its horizontal and vertical components, typically represented as A = xi + yj, where x is the horizontal component and y is the vertical component. This format allows for easier manipulation and calculation of vectors, particularly when performing operations like addition or subtraction.

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, if vector B is multiplied by a scalar k, the new vector kB will have components (kBx, kBy). This concept is crucial for determining vector F, as it requires multiplying vector B by -4 before performing the subtraction.