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

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To find the magnitude of a vector, use the formula: \( \text{Magnitude} = \sqrt{A_x^2 + A_y^2 + A_z^2} \), where \( A_x, A_y, \) and \( A_z \) are the components of the vector.

For vector A = -2i + 3j + 4k, substitute the components into the formula: \( \text{Magnitude of A} = \sqrt{(-2)^2 + 3^2 + 4^2} \).

Calculate the squares of each component: \((-2)^2 = 4\), \(3^2 = 9\), and \(4^2 = 16\).

Add the squared values: \(4 + 9 + 16 = 29\).

Take the square root of the sum to find the magnitude of vector A: \( \sqrt{29} \). Repeat the process for vector B = 3.00î + 1.00ĵ − 3.00k using the same formula.

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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 helps in determining the size of the vector irrespective of its direction.

Vector Components

Vector components are the projections of a vector along the coordinate axes, typically expressed in terms of i, j, and k unit vectors in three-dimensional space. For example, a vector A = ai + bj + ck has components a, b, and c along the x, y, and z axes respectively. Understanding components is crucial for vector addition, subtraction, and magnitude calculation.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction along the axes in a coordinate system. In three-dimensional space, the unit vectors are î, ĵ, and k̂, representing the x, y, and z directions respectively. They are essential for expressing vectors in component form and performing vector operations.