Given two vectors A = -2i + 3j + 4k and B = 3.00î+1.00ĵ − 3.00k (a) find the magnitude of each vector;

Verified step by step guidance

1. Step 1: Calculate the magnitude of vector A. The magnitude of a vector \( \vec{A} = a_i \hat{i} + a_j \hat{j} + a_k \hat{k} \) is given by the formula \( |\vec{A}| = \sqrt{a_i^2 + a_j^2 + a_k^2} \). Substitute the components of vector A into this formula.
2. Step 2: Calculate the magnitude of vector B. Use the same formula as in Step 1, substituting the components of vector B.
3. Step 3: Simplify the expressions obtained in Step 1 and Step 2 to find the magnitudes of vectors A and B respectively.
4. Step 4: Ensure all calculations are correct by double-checking the components of each vector and the arithmetic operations performed.
5. Step 5: Interpret the results to understand the lengths of vectors A and B in the context of the problem.

Key Concepts

Vector Magnitude

The magnitude of a vector is a measure of its length in space, calculated using the formula √(x² + y² + z²) for a three-dimensional vector represented as A = xi + yj + zk. This value indicates how far the vector extends from the origin to its endpoint.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction. They are often represented in the form of i, j, and k for the x, y, and z axes, respectively. Understanding unit vectors is essential for breaking down vectors into their components and calculating their magnitudes.

Vector Components

Vectors can be expressed in terms of their components along the coordinate axes. For example, a vector A = xi + yj + zk has components x, y, and z corresponding to the i, j, and k directions. This decomposition is crucial for performing operations like addition, subtraction, and magnitude calculation.