how to find magnitude of a vector

1
Determine the components of the vector. Every vector can be numerically represented in the Cartesian coordinate system with a horizontal (x-axis) and vertical (y-axis) component.^[2] It is written as an ordered pair $v=<x,y>$ .
- For example, the vector above has a horizontal component of 3 and a vertical component of -5, therefore the ordered pair is <3, -5>.
2

Draw a vector triangle. When you draw the horizontal and vertical components, you end up with a right triangle. The magnitude of the vector is equal to the hypotenuse of the triangle so you can use the Pythagorean theorem to calculate it.^[3]
3
Rearrange the Pythagorean theorem to calculate the magnitude. The Pythagorean theorem is A² + B² = C². "A" and "B" are the horizontal and vertical components of the triangle while "C" is the hypotenuse. Since the vector is the hypotenuse you want to solve for "C".
- x² + y² = v²
- v = √(x² + y²))
4
Solve for the magnitude. Using the equation above, you can plug in the numbers of the ordered pair of the vector to solve for the magnitude.^[4]
- For example, v = √((3²+(-5)²))
- v =√(9 + 25) = √34 = 5.831
- Don't worry if your answer is not a whole number. Vector magnitudes can be decimals.

1
Determine the components of both points of the vector. Every vector can be numerically represented in the Cartesian coordinate system with a horizontal (x-axis) and vertical (y-axis) component.^[5] It is written as an ordered pair $v=<x,y>$ . If you are given a vector that is placed away from the origin of the Cartesian coordinate system, you must define the components of both points of the vector.
- For example, the vector AB has an ordered pair for point A and point B.
- Point A has a horizontal component of 5 and a vertical component of 1, so the ordered pair is <5, 1>.
- Point B has a horizontal component of 1 and a vertical component of 2, so the ordered pair is <1, 2>.
2
Use a modified formula to solve for the magnitude. Because you now have two points you are dealing with, you must subtract the x and y components of each point before you solve using the equation v = √((x₂-x₁)² +(y₂-y₁)²).
- Point A is ordered pair 1 <x₁, y₁> and point B is ordered pair 2 <x₂, y₂>
3
Solve for the magnitude. Plug in the numbers of your ordered pairs and calculate the magnitude. Using our above example the calculation looks like this:^[6]
- 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 your answer is not a whole number. Vector magnitudes can be decimals.

Add New Question

Question

The coordinates of head and tail of a vector are (2, 1, 0), (-4, 2, -3). What is the magnitude of the vector?

You can use the same formula: |→a| = √((x2 – x1)^2 + (y2 – y1)^2), but add on (z2 – z1)^2 at the end for the third set of coordinates!
Question

How do I find the direction?

Use this formula: tan(y component / x component). If the vector is in quadrant 3 or 4, add a half rotation.
Question

What is the magnitude of the resultant vector of vector A-12.66 and vector B-11.93?

To get the magnitude you need to square both the vectors' magnitude and then take the underroot.
Question

How do I find the magnitude of a vertical and horizontal component if a vector is shown in a scale diagram?

If you're given an angle, use that angle and the vector's magnitude to calculate. Vx= (vector's mag)*cos(angle), Vy= (vector's mag)*sin(angle).
Question

How can I find the magnitude of vectors if there is no coordinates, an angle, and then a force?

If you are given a value for work, you can divide that value by the magnitude of the force multiplied by the cosine of the angle, since W=|F||d|cos⍺ (|d|=W/(|F|cos⍺)).
Question

How do I find the vector when only its modulus is given?

One modulus can apply to more than one vector, so any coordinates that fit the formula |a|= square root of (x^2 = y^2) should work, where |a|= the magnitude/modulus.
Question

How do I find the angle between two vectors?

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About This Article

Article SummaryX

To find the magnitude of a vector, first determine its horizontal and vertical components on their respective number lines around the origin. Next, draw the horizontal and vertical components to plot the point where they intersect. Then draw a line from the origin to that point, creating a vector triangle, which is a right triangle. Finally, use the Pythagorean theorem to calculate the triangle's hypotenuse, which is the same as the vector's magnitude. For more information on finding the magnitude of a vector, including using a modified formula when the vector is away from the origin, scroll down!

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