How to Calculate the magnitude of a vector

A vector is a geometric element with magnitude and direction. ^{[1] X Research Source} The magnitude of a vector is its length, and the direction of the vector indicates its direction. To calculate the magnitude of a vector we just need to do some simple math. In addition, we can also perform addition or subtraction of two vectors, find the angle between two vectors as well as calculate the directed product of two vectors.

Table of Contents

Find the magnitude of a vector with origin at point O

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Determine the composition of the vector. Each vector can be represented on the Oxy coordinate system (Cartesian coordinate system) along the horizontal (x-axis) and vertical (y-axis). ^{[2] X Research Source} When writing vector coordinates, the x and y coordinates are written in order

v

=<

x

,

y

>

{displaystyle v=<x,y>}

v=<x,y>

.

For example, the vector in the figure has point coordinates on the horizontal axis of 3 and coordinates on the vertical axis of -5, so we write the coordinates of this vector as <3, -5>.

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Draw a vector triangle. From the end of the vector, lower the perpendicular to the vertical and horizontal axes, we will get two equal right triangles. The magnitude of the vector in question is the length of the hypotenuse of this triangle, so we only need to apply the Pythagorean theorem to calculate its value.

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Rearrange the Pythagorean theorem to calculate the length. Pythagorean theorem: A ² + B ² = C ² . Where “A” and “B” are the horizontal and vertical coordinates of the triangle, “C” is the hypotenuse of the triangle. Since the vector under consideration is also the hypotenuse “C”, so we need to find “C”.

x ² + y ² = v ²
v = (x ² + y ² ))

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Solve the equation to find the magnitude of the vector. Substituting the values into the corresponding quantities and solving the equation will calculate the magnitude of the vector in question.

For example, v = ((3 ² +(-5) ² ))
v =√(9 + 25) = 34 = 5,831
The vector can be a decimal, so don’t worry if the calculated result is not an integer.

Calculate the magnitude of the vector outside the origin

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Determine the start and end points of the vector. All vectors can be represented on the Cartesian coordinate system as coordinates relative to the horizontal (x-axis) and vertical (y-axis) axes. ^{[3] X Research Source} The coordinates of each point will be written in pairs of x, y as follows:

v

=<

x

,

y

>

{displaystyle v=<x,y>}

v=<x,y>

. If the problem says that the vector is not on the coordinate axes in the Cartesian coordinate system, we need to determine the coordinates of the starting and ending points of that vector.

For example, the vector AB is written in pairs and in the order of point A then point B.
Point A has horizontal coordinates of 5 and vertical coordinates of 1, so point A coordinates are <5,1>.
Point B has horizontal coordinates of 1 and vertical coordinates of 2, so the coordinates of point B are <1,2>.

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Use the transformed formula to calculate the magnitude of the vector. Now that we have the coordinates of the beginning and the end of the vector, we need to get the difference of the x and y coordinates of these two points, then apply the formula v = √((x ₂ -x ₁ ) ² +(y ₂ -y ₁ ) ² ). ^{[4] X Research Sources}

Where <x ₁ , y ₁ > are the coordinates of point A, point B has a coordinate pair <x ₂ , y ₂ >.

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Solve the equation. Assign the corresponding x, y values to the formula and solve the equation to get the magnitude of the vector. Using the above example, we can calculate the following:

v = ((x ₂ -x ₁ ) ² +(y ₂ -y ₁ ) ² )
v = ((1-5) ² +(2-1) ² )
v = ((-4) ² +(1) ² )
v = (16+1) = (17) = 4.12
Since the magnitude of the vector can be a decimal, don’t worry if the calculated result is not an integer.

Steps

Find the magnitude of a vector with origin at point O

Calculate the magnitude of the vector outside the origin

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