# How to Add or Subtract Vectors

Vectors are physical quantities that consist of a magnitude as well as a direction (for example velocity, acceleration, and displacement), as opposed to scalars, which consist of magnitude only (for example speed, distance, or energy). While scalars can be added by adding their magnitudes (for example 5 kJ of work plus 6kJ of work equals 11kJ of work), vectors are slightly more complicated to add or subtract. See Step 1 below to learn several ways to tackle vector addition and subtraction.

### Method 1 Adding and Subtracting Vectors With Known Components

1. 1
Express a vector's dimensional components via vector notation. Because vectors have magnitude and direction, it's usually possible to break them into pieces based on their x, y, and/or z dimensionality. These dimensions 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 as simple as adding or subtracting their x, y, and z coordinates.
• 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. Our example below deals with 3-dimensional vectors, but the process is analogous for 2-D or 1-D vectors.
• Let's say that we have two 3-dimensional vectors, vector A and vector B. We might write these vectors in vector notation as A = <a1, b1, c1> and B = <a2, b2, c2>, where a1 and a2 are their x components, b1 and b2 are their y components, and c1 and c2 are their z components.
2. 2
To add two vectors, add their components. If two vectors' components are known, it's possible to add the vectors by adding their corresponding dimensional components. In other words, add the x component of the first vector to the x component of the second and do the same 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 = <a1+a2,b1+b2,c1+c2>.
• Let's add two vectors A and B. A = <5, 9, -10> and B = <17, -3, -2>. A + B = <5+17, 9+-3, -10+-2>, or <22, 6, -12>.
3. 3
To subtract two vectors, subtract their components. As we'll discuss later, subtracting one vector from another can be thought of adding its "reverse". If the components of two vectors are known, subtracting one vector from another can simply be done by subtracting the components of the first from the second (or by adding their negatives).
• In general terms, A-B = <a1-a2,b1-b2,c1-c2>
• 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 <28, -4, 13>.

### Method 2 Adding and Subtracting Visually Using the Head to Tail Method

1. 1
Represent vectors visually by drawing them with a head and tail. Since vectors have magnitude and direction, they can be said to have a tail and a head. In other words, vectors can be said to have a "beginning point" and an "end point" pointing in the direction of the vector whose distance from the beginning point is equal to the magnitude of the vector. When drawn visually, vectors take the form of arrows. The "point" of the arrow is the vector's head and the "base" of the arrow is the tail.
• If you are making a scale drawing of a vector, you must take care to measure and draw all angles accurately. Mis-drawn angles will be reflected in the resultant answer when two vectors are added or subtracted with the method in this section.
2. 2
To add, draw or move the second vector so that its tail meets the first's head. 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.
• Note that the order you join the vectors in is not important assuming you use the same starting point. Vector A + Vector B = Vector B + Vector A
3. 3
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.
4. 4
If adding or subtracting more than two vectors, join all other vectors head-to-tail in sequence. The order in which you join the vectors does not matter. This method can be used for any number of vectors.
5. 5
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 added vectors (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/or z components of all the vectors.
• 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.
6. 6
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".

### Method 3 Adding and Subtracting Vectors by Finding Their Dimensional Components

1. 1
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. Assuming a 2-D vector, first, set 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).
• 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
2. 2
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>.
3. 3
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=√60.73 = 7.79
4. 4
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
5. 5
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".

## Tips

• Vectors represented in the form xi + yj + zk can be added or subtracted by simply adding or subtracting to coefficients of the three unit vectors. The answer will also be in i,j,k form.
• Column vectors can be added or subtracted by simply adding or subtracting the values in each row.
• Vectors are not to be confused with magnitudes.
• Vectors in the same direction can be added or subtracted by adding or subtracting their magnitudes. If you add two vectors in opposite directions, their magnitudes are subtracted, not added.
• You can find the magnitude of a vector in three dimensions by using the formula a2=b2+c2+d2, where a is the magnitude of the vector, and b, c, and d are the components in each direction.

## Article Info

Categories: Coordinate Geometry | Algebra