Calculating the Magnitude and Direction of a Vector: A Comprehensive Guide

Understanding vectors is a fundamental concept in mathematics and physics. This article will guide you through the process of finding the magnitude and direction of a vector using its component values along the X-axis and Y-axis. We will explore the application of the Pythagorean theorem and trigonometry to solve this problem.

What is a Vector?

A vector is a mathematical object that possesses both magnitude (size) and direction. Vectors are commonly represented as arrows, with the length of the arrow indicating the magnitude and the arrowhead indicating the direction. In two-dimensional space, a vector is defined by its horizontal (X-axis) and vertical (Y-axis) components.

Given Data and Problem Statement

The problem presented involves a vector with a component along the X-axis equal to 25 units and a component along the Y-axis equal to 60 units. Our goal is to determine the magnitude and direction of this vector.

Calculating the Magnitude of the Vector

The magnitude of a vector can be calculated using the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In the context of vectors, the magnitude (V) is given by:

Magnitude

[ V sqrt{V_x^2 V_y^2} ]

where (V_x) is the component along the X-axis and (V_y) is the component along the Y-axis.

Calculation

Given:

 V_x  25 text{ units} V_y  60 text{ units}

Calculating the magnitude:

[ V sqrt{25^2 60^2} sqrt{625 3600} sqrt{4225} 65 text{ units} ]

Calculating the Direction of the Vector

The direction of the vector can be determined using the arctangent function (inverse tangent). The direction angle (theta) is found using the formula:

Direction

[ theta tan^{-1} left( frac{V_y}{V_x} right) ]

Substitute the given values to find the angle:

[ theta tan^{-1} left( frac{60}{25} right) tan^{-1} 2.4 ]

Using a scientific calculator or trigonometric tables, we find:

[ theta approx 67.38^circ ]

Summary

Magnitude: 65 units

Direction: Approximately 67.38 degrees from the positive X-axis.

Visualization

The vector was drawn with x and y components as shown in the attached picture. The vector (V) has x 25 units and y 60 units. Using the Pythagorean theorem, the magnitude of (V) is determined to be 65 units. The direction is calculated to be 67.38 degrees from the x-axis.

Solving for the Magnitude of the Vector

To find the magnitude:

[ V^2 x^2 y^2 ]

[ V^2 25^2 60^2 ]

[ V^2 625 3600 ]

[ V^2 4225 ]

[ V sqrt{4225} 65 text{ units} ]

Solving for the Angle

To find the angle (theta):

[ tan theta frac{y}{x} ]

[ tan theta frac{60}{25} ]

[ tan theta 2.4 ]

[ theta tan^{-1}(2.4) approx 67.38^circ ]

Conclusion

Understanding the magnitude and direction of a vector is crucial in various applications, from engineering to physics. By using the Pythagorean theorem and trigonometric functions, we can accurately determine these properties.