How to Add or Subtract Vectors

Many common physical quantities are often vectors or scalars. Vectors are akin to arrows and consist of a positive magnitude (length) and importantly a direction. on the other hand scalars are just numerical values sometimes possibly negative. Note that although vector magnitudes are positive or perhaps zero the components of vectors can of course be negative indicating vector directed contrary to the coordinate or reference direction.

Examples of vectors: force, velocity, acceleration, displacement, weight, magnetic field, etc. Examples of scalars: mass, temperature, speed, distance, energy, voltage, electric charge, pressure within a fluid, etc.

While scalars can be added directly like numbers (e.g. 5 kJ of work plus 6kJ equals 11kJ ; or 9 volt plus minus 3 volt gives 6 volt: +9v plus -3v gives +6v ), vectors are slightly more complicated to add or subtract, although collinear vectors are easy and behave like adding numbers which may be negative. See below several ways to tackle vector addition and subtraction.

Things You Should Know

Similarly, subtract vectors by subtracting each of their matching components.

To add vectors visually, you'll need to draw them from head to tail so that the second vector's tail meets the first's vector's head.

When subtracting vectors visually, reverse the direction of the vector but keep its magnitude the same.

Adding and Subtracting Vectors With Known Components

$A=< a1, b1, c1, > B=< a2, b2, c2, >$

Express a vector in terms of components in some coordinate system usually x, y, and possibly z in usual 2 or 3 dimensional space (higher dimensionality is possible too in some mathematical situations).

These component parts are usually expressed with a notation similar to that used to describe points in a coordinate system (e.g. <x,y,z>, etc.). If these pieces are known, adding or subtracting vectors is just a simple adding or subtracting the x, y, and z components.

Note that vectors can be 1, 2, or 3-dimensional. Thus, vectors can have an x component, an x and y component, or an x, y, and z component.

Let's say that we have two 3-dimensional vectors, vector A and vector B. We might write these vectors in components as A = <Ax,Ay,Az> and B = <Bx,By,Bz>, using x y z components accordingly.

$A+B= < ( a1 + a2 ) , ( b1 + b2 ) , ( c1 + c2 ) > example: A= < 5, 9, -10, > B= < 17, -3, -2, > A+B= < ( 5+17 ) , ( 9+(-3) ) , ( (-10)+(-2) ) > A+B = < 22, 6, -12, >$

In other words, add the x component of the first vector to the x component of the second and so on for y and z.

The answers you get from adding the x, y, and z components of your original vectors are the x, y, and z components of your new vector.

In general terms, A+B = <Ax+Bx,Ay+By,Az+Bz>.

Let's add two vectors A and B. Example: A = <5, 9, -10> and B = <17, -3, -2>. A + B = <5+17, 9+-3, -10+-2>, or <22,6, -12>.

$A-B= < ( a1 - a2 ) , ( b1 - b2 ) , ( c1 - c2 ) > example: A= < 18, 5, 3, > B= < -10, 9, -10 > A-B= < ( 18-(-10) ) , ( 5-9 ) , ( 3-(-10) ) > A-B = < 28, -4, 13 >$

To subtract two vectors, subtract their components.

Note that subtracting one vector from another A-B can be thought of adding the "reverse" of that second A+(-B).

In general terms, A-B = <Ax-Bx,Ay-By,Az-Bz>

Let's subtract two vectors A and B. A = <18, 5, 3> and B = <10, 9, -10>. A - B = <18-10, 5-9, 3-(-10)>, or <8, -4, 13>.

Represent vectors visually by drawing them with a head and tail.

Since vectors have magnitude and direction, they are likened to arrows with a tail and a head and a length. Vectors can be said to have a "beginning point" and an "end point". The "sharp point" of the arrow is the vector's head and the "base" of the arrow is the tail.

When making a scale drawing of a vector, you must take care to measure and draw all angles accurately. Mis-drawn angles will lead to poor answers.

To add 2 vectors, draw the second vector B so that its tail meets the head of the first A.

This is referred to as joining your vectors "head to tail". If you are only adding two vectors, this is all you'll need to do before finding your resultant vector A+B. Vector B may need to be slid into position without altering its orientation, called parallel transport.

Note that the order you join the vectors in is not important. Vector A + Vector B = Vector B + Vector A

To add 2 vectors, draw the second vector B so that its tail meets the head of the first A. This is referred to as joining your vectors "head to tail". If you are only adding two vectors, this is all you'll need to do before finding your resultant vector A+B. Vector B may need to be slid into position without altering its orientation, called parallel transport.

Note that the order you join the vectors in is not important. Vector A + Vector B = Vector B + Vector A

To subtract, add the "negative" of the vector.

Subtracting vectors visually is fairly simple. Simply reverse the vector's direction but keep its magnitude the same and add it to your vector head to tail as you would normally. In other words, to subtract a vector, turn the vector 180o around and add it.

If adding or subtracting more than two vectors, join all other vectors head-to-tail in sequence.

Actually the order in which you join the vectors does not matter. This method can be used for any number of vectors.

To get the result:

Draw a new vector from tail of the first vector to the head of the last. Whether you are adding/subtracting two vectors or a hundred, the vector stretching from the original starting point (the tail of your first vector) to end point of your final added vector (the head of your last vector) is the resultant vector, or the sum of all your vectors. Note that this vector is identical to the vector obtained by adding the x,y, and perhaps z components of all the vectors separately.

If you drew all of your vectors to scale, measuring all angles exactly, you can find the magnitude of the resultant vector by measuring its length. You can also measure the angle that the resultant makes with either a specified vector or the horizontal/vertical etc. to find its direction.

If you didn't draw all vectors to scale, you probably need to calculate the magnitude of the resultant using trigonometry. You may find the Sine Rule and the Cosine Rule helpful here. If you are adding more than two vectors together, it is helpful to first add two, then add their resultant with the third vector, and so on. See the following section for more information.

$\stackrel{\to }{A}+\stackrel{\to }{B}=\stackrel{\to }{R}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{R}\text{has a velocity of x m/s at y° to the horizontal}$

Represent your resultant vector via its magnitude and direction.

Vectors are defined by their length and direction. As noted above, assuming you drew your vectors accurately, your new vector's magnitude is its length and its direction is its angle relative to the vertical, horizontal, etc. Use the units of your added or subtracted vectors to choose the units for your resultant vector's magnitude.

For example, if the vectors we added represented velocities in ms-1, we might define our resultant vector as "a velocity of x ms-1 at yo to the horizontal".

Adding and Subtracting Vectors by Finding Components

Use trigonometry to find a vector's components.

To find a vector's components, it's usually necessary to know its magnitude and its direction relative to the horizontal or vertical and to have a working knowledge of trigonometry. Taking a 2-D vector first: set or imagine your vector as the hypotenuse of a right triangle whose other two sides are parallel to the x and y axes. These two sides can be thought of as head-to-tail component vectors that add to create your original vector.

The lengths of the two sides are equal to the magnitudes of the x and y components of your vector and may be calculated using trigonometry. If x is the magnitude of the vector, the side adjacent to the vector's angle (relative to the horizontal, vertical, etc.) angle is xcos(θ), while the side opposite is xsin(θ).

$X=3\phantom{\rule{10px}{0ex}}\theta =135°\phantom{\rule{0ex}{0ex}}X=XCos\left(\theta \right)\phantom{\rule{10px}{0ex}}Y=XSin\left(\theta \right)\phantom{\rule{0ex}{0ex}}X=3cos\left(135°\right)\phantom{\rule{10px}{0ex}}Y=3Xsin\left(135°\right)\phantom{\rule{0ex}{0ex}}X=-2.12\phantom{\rule{10px}{0ex}}Y=2.12$

It's also important to note the direction of your components. If the component points in the negative direction of one of your axes, it is given a negative sign. For example, in a 2-D plane, if a component points to the left or downwards, it is given a negative sign.

For example, let's say that we have a vector with a magnitude of 3 and a direction of 135o relative to the horizontal. With this information, we can determine that its x component is 3cos(135) = -2.12 and its y component is 3sin(135) = 2.12

A = <X1,Y1>
B = <X2,Y2>
A + B = <(X1 + X2),(Y1 + Y2)>
Example:
A + B = <(-2.12 + 5.78),(2.12 + (-9))>
A + B = <3.66,-6.88>

Add or subtract two or more vectors' corresponding components.

When you've found the components of all of your vectors, simply add their magnitudes together to find the components of your resultant vector. First, add all the magnitudes of the horizontal components (those parallel to the x-axis) together. Separately, add all the magnitudes of the vertical components (those parallel to the y-axis). If a component has a negative sign (-), its magnitude is subtracted, rather than added. The answers you obtain are the components of your resultant vector.

For instance, let's say that our vector from the previous step, <-2.12, 2.12>, is being added to the vector <5.78, -9>. In this case, our resultant vector would be <-2.12+5.78, 2.12-9>, or <3.66, -6.88>.

c2=a2+b2

a = 3.66
b = -6.88

c2 = (3.662) + (-6.88)2
c2 = 13.40 + 47.33
c = $\sqrt{60.73}$
c = 7.99

Calculate the magnitude of the resultant vector using the Pythagorean Theorem.

The Pythagorean Theorem, c2=a2+b2, solves for the side lengths of right triangles. Since the triangle formed by our resultant vector and its components is a right triangle, we can use it to find our vector's length and therefore its magnitude. With c as the magnitude of the resultant vector, which you're solving for, set a as the magnitude of its x component and b as the magnitude of its y components. Solve with algebra.

To find the magnitude of the vector whose components we found in the previous step, <3.66, -6.88>, let's use the Pythagorean Theorem. Solve as follows:

c2=(3.66)2+(-6.88)2
c2=13.40+47.33
c=$\sqrt{60.73}$ = 7.79

$\theta ={tan}^{-1}\left(\frac{b}{a}\right)\phantom{\rule{0ex}{0ex}}\text{Example:}\phantom{\rule{0ex}{0ex}}\theta ={tan}^{-1}\left(\frac{-6.88}{3.66}\right)\phantom{\rule{0ex}{0ex}}\theta ={tan}^{-1}\left(-1.88\right)\phantom{\rule{0ex}{0ex}}\theta ={-61.99}^{°}$

Calculate the direction of the resultant with the tangent function.

Finally, find the resultant vector's direction. Use the formula θ=tan-1(b/a), where θ is the angle that the resultant makes with the x-axis or the horizontal, b is the magnitude of the y component, and a is the magnitude of the x component.

To find the direction of our example vector, let's use θ=tan-1(b/a).

θ=tan-1(-6.88/3.66)
θ=tan-1(-1.88)
θ=-61.99o

Resultant Vector C is a force of 7.79N @ -61.99o to the horizontal.

Represent your resultant vector via its magnitude and direction.

As noted above, vectors are defined by their magnitude and direction. Be sure to use the proper units for your vector's magnitude.

For example, if our example vector represented a force (in Newtons), then we might write it as "a force of 7.79 N at -61.99o to the horizontal".

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