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

Verified step by step guidance

1. To find the magnitude of a vector, use the formula: \( |\mathbf{A}| = \sqrt{A_x^2 + A_y^2} \), where \( A_x \) and \( A_y \) are the components of vector \( \mathbf{A} \).
2. For vector \( \mathbf{A} = 4\mathbf{i} + 7\mathbf{j} \), identify the components: \( A_x = 4 \) and \( A_y = 7 \).
3. Substitute the components of vector \( \mathbf{A} \) into the magnitude formula: \( |\mathbf{A}| = \sqrt{4^2 + 7^2} \).
4. Similarly, for vector \( \mathbf{B} = 5\mathbf{i} - 2\mathbf{j} \), identify the components: \( B_x = 5 \) and \( B_y = -2 \).
5. Substitute the components of vector \( \mathbf{B} \) into the magnitude formula: \( |\mathbf{B}| = \sqrt{5^2 + (-2)^2} \).

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.

Vector Components

Vectors are represented by components along the coordinate axes, typically denoted as i and j for the x and y axes, respectively. Understanding vector components is crucial for calculating vector magnitude and performing operations like addition or subtraction.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, which is essential for calculating vector magnitudes.