Vectors are geometric representations of physical quantities that consist of a module (length), a direction, and a sense, such as velocity, acceleration, and displacement, unlike scalar quantities, which are only represented with a Numeric value, such as speed, distance, and energy. While scalar quantities can be added by adding the various numerical values (for example, 5 kJ of work plus 6 kJ of work add up to 11 kJ of work), adding and subtracting vectors is slightly more complicated. Read this article if you want to learn different ways to add and subtract vectors.
Method 1 of 3: Add and Subtract Vectors with Known Components
Step 1. Express the dimensional components of a vector with vector notation
Since vectors have a scalar and a directional magnitude, they can usually be divided dimensionally into different parts based on their x, y and / or z coordinates. These dimensions are often expressed in a notation similar to that used to locate points in a coordinate system (for example,). If we know these components, adding or subtracting vectors is as simple as adding or subtracting their x, y, and z coordinates.
- Keep in mind that vectors can have 1, 2 or 3 dimensions. Therefore, vectors can have only one x component, the x and y components, or the x, y, and z components. The example you can see below shows three-dimensional vectors, but the process is the same for two-dimensional and one-dimensional vectors.
- Suppose we have two three-dimensional vectors A and B. We can express these vectors in vector notation as A = and B =, where a1 and a2 are their x components, b1 and b2 are their y components, and c1 and c2 are their z components.
Step 2. To add two vectors, add their components
If we know the components of two vectors, these vectors can be added 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 the y and z components. The results you get after adding the x, y, and z components of the original vectors are the x, y, and z components of the new vector.
- In general terms, A + B =.
- Let's add two vectors A and B. A = and B =. A + B =, or.
Step 3. To subtract two vectors, subtract their components
As we will see later, subtracting one vector from another can be equivalent to adding its "opposite." If we know the components of two vectors, we can subtract one vector from another by subtracting the components of the first from the second simply (or by adding their negatives).
- In general terms, A-B =
- Let vector A subtract vector B. A = and B =. A - B =, or.
Method 2 of 3: Adding and Subtracting Vectors Using the Graphical Head-to-Tail Method
Step 1. Represent the vectors graphically by drawing them with the head and tail
Since vectors have both scalar and directional magnitudes, they can be said to have a head and a tail. In other words, it can be said that a vector starts at one point and ends at another, in the direction in which the distance between the start point and the end point is equal to the scalar magnitude of said vector. When graphed, vectors are arrow-shaped. The tip of the arrow is the "head", and the base of the arrow is the "tail."
If you draw a vector to scale, be careful to measure and draw all the angles accurately. If the angles do not have the appropriate measure, the imprecision will be reflected in the result of the addition or subtraction of vectors by the graphical method
Step 2. To add, draw or shift the second vector so that its tail coincides with the head of the first
This is called "joining the head to the tail." If you simply want to add two vectors, this is the only thing you will have to do before finding the resulting vector.
Note that the order in which you join the vectors is not important, as long as you use the same starting point. Vector A + Vector B = Vector B + Vector A
Step 3. To subtract, add the "negative" of the vector
Subtracting vectors graphically is pretty straightforward. You just have to reverse the direction of the vector keeping its direction and its scalar magnitude and add it to the other head-to-tail vector as you would for any sum of vectors. In other words, to subtract one vector from another, rotate the first one 180ºor on itself and add it to the second.
Step 4. If you want to add or subtract more than two vectors, connect all the vectors head to tail consecutively
The order in which you join the vectors does not matter. This method can be used for any number of vectors.
Step 5. Draw a new vector from the tail of the first vector to the head of the last
Regardless of whether you want to add or subtract two vectors (or hundreds of them), the vector that extends from the original starting point (the tail of the first vector) to the ending point of the added vectors (the head of the last) is the resultant vector, or the sum of all vectors. Note that this vector is identical to the vector obtained by adding the x, y and z components of all the vectors.
- If you draw all vectors to scale, measuring their angles exactly, you can find the scalar magnitude of the resulting vector by measuring its length. You can also measure the angle that the resulting vector makes with a specified vector, with the horizontal or with the vertical to find its direction.
- If you don't draw all the vectors to scale, you will probably have to calculate the scalar magnitude of the resultant using trigonometry. The law of sine and the law of cosine can be helpful in this case. If you add more than two vectors, it is advisable to first add two, then add the resultant to a third vector, and so on. Read the next step for more information.
Step 6. Represent the resulting vector by its module, its direction and its sense
Vectors are defined by their modulus (length), their direction, and their sense. As we have indicated before, if you draw the vectors with precision, the scalar magnitude or modulus of the resulting vector corresponds to its length, and its direction is given by the angle it forms with the vertical, the horizontal, etc. Use the units of the added or subtracted vectors to express the magnitude of the resulting vector.
- For example, if the summed vectors represent velocities in ms-1, we can define the resulting vector as "a speed of x ms-1 to and or from the horizontal ".
Method 3 of 3: Add and Subtract Vectors by Finding Their Dimensional Components
Step 1. Use trigonometry to find the components of a vector
To find the components of a vector, it is usually necessary to know its module, its direction and its sense in relation to the horizontal or vertical, in addition to having knowledge of trigonometry. Assuming we have a two-dimensional vector, first position it as the hypotenuse of a right triangle whose legs (the other two sides) are parallel to the Y axis and the X axis. You can visualize these two sides as the head-to-tail components whose sum generates as result the original vector.
- The lengths of the two sides are equal to the modules of the x and y components of the vector and can be calculated using trigonometric laws. If "x" is the module of the vector, the side adjacent to the angle of the vector (relative to the horizontal, vertical, etc.) is xcos (θ), while the opposite side is xsin (θ).
- It is also important to take into account the direction and sense of the components. If the component points to the negative direction of one of the axes, its magnitude is expressed with a negative sign. For example, in a two-dimensional plane, if a component points left or down, it is preceded by a negative sign.
- For example, suppose we have a vector of modulus 3 and it forms an angle of 135or with the horizontal. With this information, we can determine that its x component is 3cos (135) = - 2.12 and that its component y is 3sin (135) = 2.12.
Step 2. Add or subtract the components corresponding to two or more vectors
Once you have calculated the components of all the vectors, you just have to add their magnitudes to find the components of the resulting vector. First, add the magnitudes of the horizontal components (parallel to the X axis). Separately, add all the magnitudes of the vertical components (parallel to the Y axis). If a component has a negative sign (-), its modulus is subtracted rather than added. The result you obtain will correspond to the components of the resulting vector.
For example, suppose we want to add the vector from the previous step, and the vector. In this case, the resulting vector would be, or
Step 3. Find the modulus of the resultant vector using the Pythagorean theorem
The Pythagorean theorem, c2= a2+ b2, is used to find the length of the sides of a right triangle. Since the triangle formed by the resulting vector and its components is a right triangle, we can use this theorem to find the length of the vector and, therefore, its modulus. Consider c as the modulus of the resulting vector, which you have to find, a as the modulus of the x component and b as the modulus of the y component. Solve the operation algebraically.
To find the magnitude of the vector whose components we have calculated in the previous step,, we will use the Pythagorean theorem. Solve the operation as follows:
- c = √60.73 = 7.79
Step 4. Find the direction and sense of the resulting vector with the tangential function
Finally, find the direction and sense of the resulting vector. Use the formula θ = tan-1(b / a), where θ is the angle that the resulting vector makes with the horizontal or X axis, b is the module of the y component, and a is the module of the x component.
To find the direction and sense of the vector that we have used as an example, we will use the formula θ = tan-1(b / a).
- θ = so-1(-6.88/3.66)
- θ = so-1(-1.88)
- θ = -61.99or
Step 5. Represent the resulting vector taking into account its module, its direction and its sense
As we have already indicated before, vectors are defined by their module, their direction and their sense. Be sure to use the appropriate units to express the magnitude of the vector.
- For example, if the vector in question represents a force (in newtons), we can express it as "a force of 7.79 N forming -61.99 or with the horizontal ".
- Vector quantities should not be confused with scalar quantities.
- Vectors that have the same direction and sense can be added or subtracted by adding or subtracting, respectively, their modules. Yes sums two vectors with opposite direction, their modules must subtract, do not join.
- You can find the modulus of a three-dimensional vector using the formula a2= b2+ c2+ d2, where a is the module of the vector, and b, c, and d are the different dimensional components.
- Vectors represented by the form x i + y j + z k can be added or subtracted by adding or subtracting the coefficients of each unit vector. The answer will also be expressed by the form i, j, k.
- The vectors expressed in columns can be added or subtracted by adding or subtracting the numerical values of each row.