91Ó°ÊÓ

The Pythagorean theorem is a mathematical principle that helps us find the magnitude (or length) of a vector in two-dimensional space. It is based on the relationship between the sides of a right triangle. The theorem states that, in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
For vectors, the hypotenuse represents the vector itself. The sides of the triangle parallel to the axes represent the vector's components: the x-component and the y-component. This theorem helps us calculate the magnitude of a vector as follows:

• Square each of the vector components.
• Add these squares together.
• Take the square root of the result.

In our example, the x-component is \(-25.0\) units and the y-component is \(40.0\) units. Therefore, the magnitude of the vector is calculated as\[|V| = \sqrt{(-25)^2 + (40)^2} = \sqrt{625 + 1600} = \sqrt{2225} \approx 47.17\text{ units}.\] The Pythagorean theorem simplifies finding vector magnitudes, ensuring an understanding of vector lengths in a plane.

The arctangent function, represented by \(\arctan\), is used to determine the angle between the vector and the positive x-axis in our coordinate plane.
It calculates the angle from the tangent of the opposite and adjacent sides of a right triangle, which correspond to the y-component and x-component of a vector, respectively.
We use the formula:\[\theta = \arctan\left(\frac{V_y}{V_x}\right)\]For our vector with \(V_y = 40\) and \(V_x = -25\),\[\theta = \arctan\left(\frac{40}{-25}\right).\]On a calculator, this returns an angle of \(-58.01°\).Keep in mind that the result directly from the arctangent function might not always correctly represent the vector's direction in its particular quadrant. Hence, further analysis is necessary to determine the exact orientation.

Quadrant analysis is crucial for adjusting the angle determination to ensure it aligns with where the vector actually lies in the coordinate plane.
A plane is divided into four quadrants, and understanding where your vector sits helps determine the precise direction of the vector.

• The first quadrant is where both x and y are positive.
• The second quadrant, where our vector resides, has a negative x and a positive y.
• In the third quadrant, both x and y are negative.
• The fourth quadrant has a positive x and negative y.

In our original calculation, the arctangent provides an angle of \(-58.01°\). However, since our vector's x-component is negative and y-component is positive, it actually resides in the second quadrant.
In order to place the angle correctly in this quadrant, we need to add \(180°\) to the arctangent output:\[121.99° = -58.01° + 180°\]The adjustment ensures the vector's true direction is noted, giving a complete picture of its orientation and magnitude as \(121.99°\) from the positive x-axis, accurately fading into the standard coordinate system view.