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4.12 C) Adding & Subtracting Vectors
4.12 C) Adding & Subtracting Vectors
We are able to add two vectors on a diagram by drawing one of the vectors, and then drawing the next vector starting from the end of the first vector. The vector between the start of the first vector and the end of the last vector (second vector in this case) will be the vector for the two vectors added together.
Example 1
The vectors z and q are given in column vector form below:
Example 1
The vectors z and q are given in column vector form below:
What is z + q?
The first step in answering this question is to draw out one the vectors. As we are adding these vectors, it does not matter which vector I choose to draw first. I am going to draw vector z first. Vector z moves 3 to the right and 2 up.
The first step in answering this question is to draw out one the vectors. As we are adding these vectors, it does not matter which vector I choose to draw first. I am going to draw vector z first. Vector z moves 3 to the right and 2 up.
The next step is to draw vector q starting from the end of the first vector. Vector q moves 4 to the right and 3 down.
We now have the start and end point of the vector that we are trying to find, which means that we are able to find the vector that we are looking for. We find the vector that we are looking for by finding the displacement in both the x and y direction. This vector moves 7 in the right direction and 1 down. This means that the x component of the column vector will be 7 and the y component of the column vector will be -1. The working and column vector are shown below:
The above method got us to the correct answer. However, it is a fairly time-consuming method because you have to draw the vectors out on a grid. A better method is to add your column vectors straight across; we add all of the x components, and we add all of the y components. Let’s prove that we obtain the same answer as earlier when we use this method.
You can see from the working that we obtain the same answer as before.
Additional Points
This method also works for subtracting vectors; we work out the answer by subtracting the x components, and then subtracting the y components.
Sometimes the question will have scalar vectors (vectors that have been multiplied by a number). The first step in answering these types of questions is to multiply out the scalar vector, which we do by multiplying the scalar number (the number on the outside) by both the x and y components of the column vector. After we have found the scalar vector, we then complete the sum/subtraction in the same way as described earlier.
This method also works for subtracting vectors; we work out the answer by subtracting the x components, and then subtracting the y components.
Sometimes the question will have scalar vectors (vectors that have been multiplied by a number). The first step in answering these types of questions is to multiply out the scalar vector, which we do by multiplying the scalar number (the number on the outside) by both the x and y components of the column vector. After we have found the scalar vector, we then complete the sum/subtraction in the same way as described earlier.
Example 2
The column vectors for a and b are shown below.
The column vectors for a and b are shown below.
Find the vector for:
The first step in answering this question is to place the vectors in their appropriate place.
The next step is to add straight across; we add both of the x components together, and we add both of the y components together. The working and answer is shown below.
Example 3
The column vectors for c and d are shown below.
The column vectors for c and d are shown below.
Find the vector for:
The first step in answering this question is to sub the vectors into the calculation.
The first part of the calculation is a scalar vector; it is 3c. The next step is to get rid of this scalar vector and we do this by multiplying the x and y components of the column vector by the scalar value that is on the outside of the vector. Therefore, we multiply both components in the column vector by 3.
We are now ready to complete the calculation, which is a subtraction calculation. We complete the subtraction calculation for the x and y components seperately. When we are doing these calculations, we need to be really careful with the signs (remember the taking of a negative number results in you adding the number; – -5 becomes + 5).
Example 4
The column vectors for e and f are shown below:
The column vectors for e and f are shown below:
Find the vector for:
The first step is to sub the vectors into the calculation.
Both of the vectors are scalar vectors; we have 4c and 5d. This means that we will want to multiply both of these vectors out. We multiply the vectors out by multiplying the number on the outside (the scalar value) by both the x and y components of the vectors.
We are now ready to complete the calculation. Like the question before, we need to be really careful with the signs.