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

검증된 단계별 안내

Step 1: Start by calculating the vector F using the formula \( F = A - 4B \). Substitute the given vectors \( A = 4\mathbf{i} - 2\mathbf{j} \) and \( B = -3\mathbf{i} + 5\mathbf{j} \) into the equation.

Step 2: Multiply the vector \( B \) by 4. This means scaling each component of \( B \) by 4: \( 4B = 4(-3\mathbf{i} + 5\mathbf{j}) = -12\mathbf{i} + 20\mathbf{j} \).

Step 3: Subtract \( 4B \) from \( A \) component-wise. For the \( \mathbf{i} \)-component: \( 4 - (-12) = 4 + 12 = 16 \). For the \( \mathbf{j} \)-component: \( -2 - 20 = -22 \). Thus, \( F = 16\mathbf{i} - 22\mathbf{j} \).

Step 4: Calculate the magnitude of \( F \) using the formula \( |F| = \sqrt{(F_x)^2 + (F_y)^2} \), where \( F_x = 16 \) and \( F_y = -22 \). Substitute these values into the formula: \( |F| = \sqrt{16^2 + (-22)^2} \).

Step 5: Determine the direction of \( F \) by calculating the angle \( \theta \) it makes with the positive \( \mathbf{i} \)-axis. Use the formula \( \theta = \arctan\left(\frac{F_y}{F_x}\right) \), where \( F_x = 16 \) and \( F_y = -22 \). Substitute these values into the formula: \( \theta = \arctan\left(\frac{-22}{16}\right) \).

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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 to find a resultant vector. In this case, vector F is calculated by subtracting a scaled version of vector B from vector A. This operation follows the rules of vector arithmetic, where corresponding components of the vectors are added or subtracted.

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 + bj, the magnitude is given by |A| = √(a² + b²). This concept is essential for determining how strong or large a vector is in a given direction.

Direction of a Vector

The direction of a vector indicates the orientation in which it acts and is often expressed as an angle or a unit vector. For a vector in the form A = ai + bj, the direction can be found using the arctangent function, θ = arctan(b/a). Understanding direction is crucial for interpreting the vector's effect in physical contexts.