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How to Calculate Magnitude of a Vector

Any vector quantity can be represented by an arrow. The length of the arrow corresponds to the order of magnitude of the vector. To determine the direction of the vector, it ‘s besides significant to define a organize system. typically, a standard coordinate system where the x-axis points to the right and the y-axis points up is used, and then the steering of the vector is given as an angle from either the ten or y-axis. The points where the vector begins and ends can besides be given in terms of their adam and yttrium coordinates. The starting period has coordinates of ( x0, y0 ), and the ending point ( where the point of the arrow is located ) has coordinates of ( x1, y1 ). Since the magnitude of a vector corresponds to its length, the magnitude is equal to the distance between these two points .

The order of magnitude of a vector corresponds to the length of the arrow.

An image showing an arrow pointing up and to the right.

The Magnitude of a Vector Formula

To find the distance between the start and ending points of the vector, and consequently its magnitude, separate the vector into two parts. The adam component of the vector is the distance it goes in the x-direction ( x1-x0 ). This distance is labeled a in the figure below. The yttrium component of the vector is the distance it goes in the y-direction ( y1-y0 ). This distance is labeled b in the figure below .

Notice that this makes a right triangle. The two sides are the x and y components, and the hypotenuse is the magnitude of the vector. therefore, if the components are known, the magnitude can be found using the Pythagorean Theorem. $ $ c^ { 2 } = a^ { 2 } +b^ { 2 } $ $ $ $ speed of light = \sqrt { ( a^ { 2 } +b^ { 2 } } $ $ finally, substituting the components for a and bacillus in the equation above gives the order of magnitude of the vector : $ $ magnitude = \sqrt { ( x_ { 1 } -x_ { 0 } ) ^ { 2 } + ( y_ { 1 } -y_ { 0 } ) ^ { 2 } } $ $

How to Find the Magnitude of a Vector for Real and Complex Numbers

A complex act is any number that contains the symbol i, which is equal to { eq } \sqrt { -1 } { /eq }. i is sometimes called the complex number unit, even though it does in truth exist ! very numbers, in contrast, are numbers that do NOT include i .

Complex Numbers

A complex count can be written in the form { equivalent } a+bi { /eq }, where a and b are real numbers. complex numbers can besides be graphed on a coordinate organization known as the complex plane, in which the x-axis corresponds to the real part of the number and the y-axis corresponds to the fanciful part. In that case, the coordinates of the complex count are ( a, b ). A complex number can besides be represented as a vector by drawing an arrow from the origin to the compass point ( a, b ) .

A complex number can be represented as a vector in the complex flat .
The complex plane, showing the real (x) and imaginary (y) axes, and a complex number that is depicted as a vector.

Magnitude of a Complex Number

The order of magnitude of this vector can besides be found using the Pythagorean theorem, but since the vector starts at the origin and x0 and y0 are both zero, the mathematics is a morsel bare :
$ $ magnitude = \sqrt { a^ { 2 } +b^ { 2 } } $ $

How to Find Vector Components From Magnitude and Angle

It ‘s besides possible to find the ten and y components of a vector if the magnitude and commission are known by using the trigonometric functions sine and cosine .

The properties of a right triangle can be used to find the components of a vector
A right triangle with the opposite, adjacent, and hypotenuse sides labeled.

For any right triangle, sine and cosine are defined as follows : $ $ \sin x=\frac { opposite } { hypotenuse } $ $ $ $ \cos x=\frac { adjacent } { hypotenuse } $ $ When a vector is graphed and the guidance is given proportional to the x-axis, the magnitude of the vector corresponds to the hypotenuse of a triangle, and the adam and y components equate to the adjacent and inverse sides, respectively. consequently, the sine and cosine functions can be used to calculate the x and yttrium components of a vector if the order of magnitude ( r ) and direction ( { eq } \Theta { /eq } ) are known : $ $ x=r\cos \Theta $ $

$ $ y=r\sin \Theta $ $

The adam and yttrium components of a vector can be found from the magnitude and commission of the vector
A triangle showing the x and y components of a vector

Problems on Magnitude of a Vector

To see how to apply these equations, let ‘s look at some examples of how to calculate the magnitude and components of a vector :

Magnitude When Components are Given

A person pushes on a box with a force of 75 N in the x-direction and 25 N in the y-direction. What is the magnitude of the total pushing force? In this problem, the components of the power vector are known. To solve this problem :

  • Draw a force triangle showing the x component of the force (75 N) and the y component of the force (25 N).
  • Use the Pythagorean theorem to find the hypotenuse of the triangle, which is the magnitude of the pushing force.

$ $ F=\sqrt { ( 25 N ) ^ { 2 } + ( 75 N ) ^ { 2 } } =79 N $ $

Magnitude When Two Points are Given

A bug is crawling across a coordinate grid. It begins at a location of (1.0, 3.0) cm and ends at a location of (8.0, 5.0 cm). What is the distance between the bug’s starting and ending point? In this problem, a vector can be drawn from the starting localization to the ending location of the bug, and the magnitude of this vector will be adequate to the distance between the get down and ending points. To find this magnitude, use the outdistance formula to find the length of the line connecting the get down and ending points. $ $ magnitude = \sqrt { ( 8.0-1.0 ) ^ { 2 } + ( 5.0-3.0 ) ^ { 2 } } $ $ $ $ magnitude = \sqrt { ( 7.0 ) ^ { 2 } + ( 2.0 ) ^ { 2 } } =7.3 cm $ $

Magnitude of a Complex Number

What is the magnitude of the complex number 12-8i? In this problem, a=12 and b=-8, so the magnitude of this complex number is : $ $ F=\sqrt { ( 12 ) ^ { 2 } + ( -8 ) ^ { 2 } } =14.4 $ $

Vector Components from Magnitude and Direction

A 100 N force is applied at an angle of 30 degrees from the horizontal. What are the x and y components of this force? In this trouble, the magnitude and guidance are given, and these can be used to find the x and y components. $ $ x= ( 100 N ) \cos 30^ { \circ } =87 N $ $ $ $ y= ( 100 N ) \sin 30^ { \circ } =50 N $ $

Lesson Summary

A vector is any measure that has a direction and a magnitude, which is the numeral size of the vector. Examples of vector quantities include force, speed, and acceleration. A vector can be represented by an arrow absorb on a coordinate plane. If the two endpoints of the vector are known, the magnitude of the vector can be found by calculating the distance between the two endpoints : $ $ magnitude = \sqrt { ( x_ { 1 } -x_ { 0 } ) ^ { 2 } + ( y_ { 1 } -y_ { 0 } ) ^ { 2 } } $ $ A building complex phone number ( a+bi ) can besides be represented as a vector, and its magnitude is given by : $ $ order of magnitude = \sqrt { ( a ) ^ { 2 } + ( b ) ^ { 2 } } $ $

last, if the magnitude and direction of the vector are already known, the x and y components can be calculated : $ $ x=r\cos \Theta $ $ $ $ y=r\sin \Theta $ $

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