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. To find the magnitude of a vector, use the formula: \( |\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} \), where \( A_x, A_y, \) and \( A_z \) are the components of vector \( \mathbf{A} \).
2. For vector \( \mathbf{A} = -2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \), substitute the components into the formula: \( |\mathbf{A}| = \sqrt{(-2)^2 + 3^2 + 4^2} \).
3. Calculate the squares of each component: \( (-2)^2 = 4 \), \( 3^2 = 9 \), and \( 4^2 = 16 \).
4. Add the squared components: \( 4 + 9 + 16 \).
5. Take the square root of the sum to find the magnitude of vector \( \mathbf{A} \). Repeat the same process for vector \( \mathbf{B} = 3.00\mathbf{i} + 1.00\mathbf{j} - 3.00\mathbf{k} \) using its components.

Key Concepts

Vector Magnitude

The magnitude of a vector is a measure of its length and is calculated using the square root of the sum of the squares of its components. For a vector A = ai + bj + ck, the magnitude is given by |A| = √(a² + b² + c²). This formula is derived from the Pythagorean theorem in three-dimensional space.

Vector Components

Vectors in three-dimensional space are expressed in terms of their components along the x, y, and z axes, typically denoted as i, j, and k. Each component represents the vector's influence in that particular direction. Understanding these components is crucial for performing operations like addition, subtraction, and finding magnitudes.

Pythagorean Theorem in 3D

The Pythagorean theorem in three dimensions extends the familiar two-dimensional version to account for an additional axis. It states that the square of the length (magnitude) of a vector is equal to the sum of the squares of its orthogonal components. This principle is fundamental in calculating distances and magnitudes in three-dimensional vector spaces.