Given two vectors A = 4i + 7j and B = 5i - 2j, (a) find the magnitude of each vector;

Verified step by step guidance

1. To find the magnitude of a vector, use the formula: \( \text{magnitude} = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the components of the vector.
2. For vector A = 4i + 7j, identify the components: \( x = 4 \) and \( y = 7 \).
3. Substitute the components of vector A into the magnitude formula: \( \text{magnitude of A} = \sqrt{4^2 + 7^2} \).
4. For vector B = 5i - 2j, identify the components: \( x = 5 \) and \( y = -2 \).
5. Substitute the components of vector B into the magnitude formula: \( \text{magnitude of B} = \sqrt{5^2 + (-2)^2} \).

Key Concepts

Vector Magnitude

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector A = ai + bj, the magnitude is given by |A| = √(a² + b²). This formula helps determine the size of the vector in the coordinate plane, regardless of its direction.

Unit Vectors

Unit vectors are vectors with a magnitude of one and are used to indicate direction. In a Cartesian coordinate system, the unit vectors i and j represent the x and y directions, respectively. They are essential for expressing vectors in terms of their components along the axes.

Vector Components

Vector components are the projections of a vector along the coordinate axes. For a vector A = ai + bj, 'a' and 'b' are the components along the x and y axes, respectively. Understanding these components is crucial for performing vector operations like addition, subtraction, and finding magnitudes.