Vectors are elements that appear very frequently in solving problems related to physics. Vectors are defined with two parameters: intensity (or modulus or magnitude) and direction. The intensity represents the length of the vector, while the direction represents the direction in which it is oriented. Calculating the magnitude of a vector is a simple operation that takes just a few steps. There are other important operations that can be performed between vectors, including adding and subtracting two vectors, identifying the angle between two vectors and calculating the vector product.
Steps
Method 1 of 2: Calculate the Intensity of a Vector Starting from the Origin of the Cartesian Plane
Step 1. Determine the components of a vector
Each vector can be represented graphically in a Cartesian plane using the horizontal and vertical components (relative to the X and Y axis respectively). In this case it will be described by a pair of Cartesian coordinates v = (x, y).
For example, let's imagine that the vector in question has a horizontal component equal to 3 and a vertical component equal to -5; the pair of Cartesian coordinates will be the following (3, -5)
Step 2. Draw the vector
By representing the vector coordinates on the Cartesian plane you will get a right triangle. The intensity of the vector will be equal to the hypotenuse of the triangle obtained; therefore, to calculate it you can use the Pythagorean theorem.
Step 3. Use the Pythagorean theorem to go back to the formula useful for calculating the intensity of a vector
The Pythagorean theorem states the following: A2 + B2 = C2. "A" and "B" represent the legs of the triangle which in our case are the Cartesian coordinates of the vector (x, y), while "C" is the hypotenuse. Since the hypotenuse is exactly the graphical representation of our vector, we will have to use the basic formula of the Pythagorean theorem to find the value of "C":
- x2 + y2 = v2.
- v = √ (x2 + y2).
Step 4. Calculate the intensity of the vector
Using the equation from the previous step and the sample vector data, you can proceed to calculate its intensity.
- v = √ (32+(-5)2).
- v = √ (9 + 25) = √34 = 5.831
- Don't worry if the result isn't an integer; the intensity of a vector can be expressed by a decimal number.
Method 2 of 2: Calculate the Intensity of a Vector Far from the Origin of the Cartesian Plane
Step 1. Determine the coordinates of both points of the vector
Each vector can be represented graphically in a Cartesian plane using the horizontal and vertical components (relative to the X and Y axis respectively). When the vector originates in the origin of the axes of the Cartesian plane, it is described by a pair of Cartesian coordinates v = (x, y). Having to represent a vector far from the origin of the axes of the Cartesian plane, it will be necessary to use two points instead.
- For example, the vector AB is described by the coordinates of point A and point B.
- Point A has a horizontal component of 5 and a vertical component of 1, so the coordinate pair is (5, 1).
- Point B has a horizontal component of 1 and a vertical component of 2, so the coordinate pair is (1, 1).
Step 2. Use the modified formula to calculate the intensity of the vector in question
Since in this case the vector is represented by two points of the Cartesian plane, we must subtract the X and Y coordinates before we can use the known formula to calculate the modulus of our vector: v = √ ((x2-x1)2 + (y2-y1)2).
In our example point A is represented by the coordinates (x1, y1), while point B from the coordinates (x2, y2).
Step 3. Calculate the intensity of the vector
We substitute the coordinates of points A and B within the given formula and proceed to perform the related calculations. Using the coordinates of our example we will get the following:
- v = √ ((x2-x1)2 + (y2-y1)2)
- v = √ ((1-5)2 +(2-1)2)
- v = √ ((- 4)2 +(1)2)
- v = √ (16 + 1) = √ (17) = 4, 12
- Don't worry if the result isn't an integer; the intensity of a vector can be expressed by a decimal number.
