In Exercises 53–56, let u = -2i + 3j, v = 6i - j, w = -3i. Find each specified vector or scalar. ||u + v||² - ||u - v||²

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First, recall the formula for the magnitude squared of a vector: for any vector \( \mathbf{a} \), \( ||\mathbf{a}||^2 = \mathbf{a} \cdot \mathbf{a} \), where \( \cdot \) denotes the dot product.

Express the vectors \( \mathbf{u} + \mathbf{v} \) and \( \mathbf{u} - \mathbf{v} \) explicitly by adding and subtracting their components: \( \mathbf{u} + \mathbf{v} = (-2 + 6)\mathbf{i} + (3 - 1)\mathbf{j} \) and \( \mathbf{u} - \mathbf{v} = (-2 - 6)\mathbf{i} + (3 + 1)\mathbf{j} \).

Calculate \( ||\mathbf{u} + \mathbf{v}||^2 = (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v}) \) by taking the dot product of \( \mathbf{u} + \mathbf{v} \) with itself.

Similarly, calculate \( ||\mathbf{u} - \mathbf{v}||^2 = (\mathbf{u} - \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) \) by taking the dot product of \( \mathbf{u} - \mathbf{v} \) with itself.

Finally, subtract the two results to find \( ||\mathbf{u} + \mathbf{v}||^2 - ||\mathbf{u} - \mathbf{v}||^2 \). You can also use the identity \( ||\mathbf{u} + \mathbf{v}||^2 - ||\mathbf{u} - \mathbf{v}||^2 = 4(\mathbf{u} \cdot \mathbf{v}) \) to simplify the calculation.

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주요 개념

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

Vector Addition and Subtraction

Vector addition combines corresponding components of two vectors to form a new vector, while subtraction finds the difference between components. Understanding how to add and subtract vectors component-wise is essential for calculating expressions like u + v and u - v.

Vector Norm (Magnitude)

The norm or magnitude of a vector is the length of the vector in space, calculated as the square root of the sum of the squares of its components. For a vector a = xi + yj, ||a|| = √(x² + y²). Squared norms (||a||²) simplify calculations by avoiding the square root.

Properties of Dot Product and Norms

The dot product relates to vector magnitudes and angles, and satisfies the identity ||u + v||² = ||u||² + 2(u·v) + ||v||². Using this, the expression ||u + v||² - ||u - v||² simplifies to 4(u·v), linking vector norms and dot products for efficient computation.