A vector is a physical quantity that has both magnitude and direction (e.g. velocity, acceleration, and displacement), as opposed to a scalar which consists of only magnitude (e.g. speed, distance, or energy). If scalars can be added by adding magnitudes (e.g. 5 kJ work plus 6 kJ work equals 11 kJ work), vectors are a bit tricky to add or subtract. See Step 1 below to learn some ways to add or subtract vectors.
Step
Method 1 of 3: Adding and Subtracting Vectors whose Components are Known
Step 1. Write down the dimensional components of the vector in vector notation
Since vectors have magnitude and direction, they can usually be broken down into parts based on the x, y, and/or z dimensions. These dimensions are usually written in a similar notation to describe a point in a coordinate system (eg and others). If you know this part, adding or subtracting vectors is very easy, just add or subtract their x, y, and z coordinates.
- Notice if the dimensions of the vector are 1, 2, or 3. Thus, the vector can have components x, x and y, or x, y, and z. Our example below uses a 3-dimensional vector, but the process is like a 1- or 2-dimensional vector.
- Suppose we have two 3-dimensional vectors, vector A and vector B. We can write these vectors using vector notation such as A = and B =, where a1 and a2 are x components, b1 and b2 are y components, and c1 and c2 are components z.
Step 2. To add the two vectors, add up their components
If the two components of a vector are known, you can add the vectors by adding the components of each. In other words, add the x-component of the first vector by the x-component of the second vector, and do the same for y and z. The answer you get from adding up the x, y, and z components of those vectors is the x, y, and z components of your 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 both vectors, subtract their components
As we'll discuss later, subtracting one vector from another, can be thought of as adding up the reciprocal vectors. If the components of both vectors are known, it is possible to subtract one vector from another by subtracting the first component from the second component (or by adding the negative components of both).
- In general terms, A-B =
- Let's subtract two vectors A and B. A = and B =. A - B =, or.
Method 2 of 3: Adding and Subtracting With Pictures Using the Head and Tail Method
Step 1. Symbolize the vector by drawing it using the head and tail
Since vectors have both magnitude and direction, we can say they have a tail and a head. In other words, a vector has a starting point and an end point that indicates the direction of the vector whose distance from the starting point is equal to the magnitude of the vector. When drawn, the vector takes the form of an arrow. The end of the arrow is the head of the vector and the end of the vector line is the tail.
If you're creating a vector drawing with dimensions, you'll need to measure and draw all the corners accurately. The wrong angle of the image will affect the resultant result when two vectors are added or subtracted using this method
Step 2. To add, draw, or move the second vector so that the tail meets the head of the first vector
This is called combining head to tail vectors. If you're just adding up two vectors, here's what you need to do before finding the resultant vector.
Note that the order in which you add vectors doesn't matter, assuming you use the same starting point. Vector A + Vector B = Vector B + Veltor A
Step 3. To subtract, add a negative sign to the vector
Reducing vectors using images is very simple. Reverse the vector direction, but keep the magnitude the same and add up your vector head and tail as usual. In other words, to subtract a vector, rotate the vector 180o and add up.
Step 4. If you add or subtract more than two vectors, combine them all in a head-to-tail order
The order of merging doesn't matter. This method can be used regardless of the number of vectors.
Step 5. Draw a new vector from the tail of the first vector to the head of the last vector
Whether you're adding/subtracting two vectors or a hundred, the vector that extends from your initial starting point (the tail of the first vector) to the end point of your last vector (the head of your last vector) is the resultant vector or the sum of all your vectors. Note that this vector is exactly the same as the vector obtained by adding up all the x, y, and/or z components.
- If you draw all your vectors to size, by measuring all the angles correctly, you can determine the magnitude of the resultant vector by measuring the length. You can also measure the angle between the resultant and any vector horizontally or vertically to determine its direction.
- If you don't draw all your vectors to size, you may have to calculate the magnitude of the resultant using trigonometry. Maybe the Sine and Cosine Rules will help. If you add more than two vectors, it's helpful to add the first vector by the second, then add the resultant of the second to the third, and so on. See the following sections for more information.
Step 6. Draw your resultant vector using its magnitude and direction
A vector is defined by its length and direction. As above, assuming you drew your vector accurately, your new vector's magnitude is its length and its direction is the angle relative to the vertical or horizontal direction. Use the unit vectors that you add or subtract to determine the units for the magnitude of your resultant vector.
For example, if the added vectors represent velocity in ms-1, then the resultant vector can be defined as "speed x ms-1 against y o to the horizontal direction.
Method 3 of 3: Adding and Subtracting Vectors by Specifying Vector Dimensional Components
Step 1. Use trigonometry to determine the components of a vector
To find the components of a vector, you usually need to know its magnitude and direction relative to the horizontal or vertical direction and understand trigonometry. Assuming a 2-dimensional vector, first, think of your vector as the hypotenuse of a right triangle whose two sides are parallel to the x and y directions. These two sides can be thought of as components of a head-to-tail vector that add up to form your vector.
- The lengths of both sides are equal to the x and y components of your vector and can be calculated using trigonometry. If x is a vector magnitude, the side adjacent to the vector angle (relative to the horizontal, vertical, and other directions) is xcos(θ), while the opposite side is xsin(θ).
- It's also very important to note the direction of your components. If the component points to a negative coordinate, it is given a negative sign. For example, in a 2-dimensional plane, if a component is pointing to the left or down, it is negative.
- For example, let's say we have a vector with magnitude 3 and direction 135o relative to the horizontal. With this information, we can determine that the x component is 3cos(135) = -2, 12 and the y component is 3sin(135) = 2, 12
Step 2. Add or subtract two or more related vectors
Once you've found the components of all your vectors, add them up to find the components of your resultant vector. First, add up all the magnitudes of the horizontal components (which are parallel to the x-direction). Separately, add up all the magnitudes of the vertical components (which are parallel to the y-direction). If a component is negative (-), its magnitude is subtracted, not added. The answer you get is the component of your resultant vector.
For example, the vector from the previous step,, is added to the vector. In this case, the resultant vector becomes or
Step 3. Calculate the magnitude of the resultant vector using the Pythagorean Theorem
Pythagorean Theorem c2=a2+b2, is used to find the side lengths of a right triangle. Since the triangle formed by our resultant vector and its components is a right triangle, we can use it to find the vector's length and magnitude. With c as the magnitude of the resultant vector, which you are looking for, suppose a is the magnitude of the x component and b is the magnitude of the y component. Solve using algebra.
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To find the magnitude of the vector whose components we have been looking for in the previous step,, use the Pythagorean Theorem. Solve as follows:
- c2=(3, 66)2+(-6, 88)2
- c2=13, 40+47, 33
- c=√60, 73 = 7, 79
Step 4. Calculate the resultant direction using the Tangent function
Finally, find the resultant vector of the direction. Use the formula =tan-1(b/a), where is the size of the angle formed in the x or horizontal direction, b is the size of the y component, and a is the size of the x component.
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To find the direction of our vector, use =tan-1(b/a).
- =tan-1(-6, 88/3, 66)
- =tan-1(-1, 88)
- =-61, 99o
Step 5. Draw your resultant vector according to its magnitude and direction
As written above, vectors are defined by their magnitude and direction. Make sure to use the appropriate units for your vector size.
For example, if our vector example represents a force (in Newtons), then we can write it "force 7.79 N by -61.99 o to horizontal".
Tips
- Vector is different from big.
- Vectors with the same direction can be added or subtracted by adding or subtracting their magnitudes. If you sum up two vectors that are opposite, their magnitudes are subtracted, not added.
- Vectors represented in the form x i + y j + z k can be added or subtracted by adding or subtracting the coefficients of the three unit vectors. The answer is also in the form of i, j, and k.
- You can find the size of a three-dimensional vector using the formula a2=b2+c2+d2 where a is the magnitude of the vector, and b, c and d are the components of each direction.
- Column vectors can be added and subtracted by adding or subtracting the values of each row.
