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4.12 B) Same, Opposite & Scalar
4.12 B) Same, Opposite & Scalar
Same
Vectors are the same if they have the same magnitude and direction. For example, we have two vectors.
Vectors are the same if they have the same magnitude and direction. For example, we have two vectors.
We work out the column vector for each of the vectors by working out the displacement in both the x and y direction.
The column vectors for the two vectors above are the same as both of these vectors move 2 towards the right and 3 up. These vectors are also parallel to one another, which means that the distance between the two vectors will be the same/ the two vectors will never touch.
Opposite
Suppose that we are given one vector and are asked to find the opposite vector to this vector. We are able to do this by multiplying the vector that we have by -1. Multiplying a vector by -1 changes the signs of the components in the vector.
Example 1
Suppose that we are given the column vector labelled u as:
Suppose that we are given one vector and are asked to find the opposite vector to this vector. We are able to do this by multiplying the vector that we have by -1. Multiplying a vector by -1 changes the signs of the components in the vector.
Example 1
Suppose that we are given the column vector labelled u as:
What is the opposite vector for this?
We are able to obtain the opposite vector by multiplying the vector that we are given by -1. The working and the column vector are shown below.
We are able to obtain the opposite vector by multiplying the vector that we are given by -1. The working and the column vector are shown below.
Let’s draw these vectors. The first vector moves 1 to the right and 4 up. This is shown on the diagram below.
I am going to start the opposite vector from the end point of the first vector. The opposite vector moves 1 to the left and 4 down. The opposite and the original vector are shown on the diagram below.
From the above diagram, you can see that the two vectors are the opposite of one another.
We know that vector u and the new vector that we have obtained are opposite to one another. Another way to write the new vector is to simple write -u.
This technique is incredible useful when we are given geometry problems with vectors. We will be looking at these types of question in the last part of the whole vector section.
Example 2
We are given the following 2 points.
We know that vector u and the new vector that we have obtained are opposite to one another. Another way to write the new vector is to simple write -u.
This technique is incredible useful when we are given geometry problems with vectors. We will be looking at these types of question in the last part of the whole vector section.
Example 2
We are given the following 2 points.
The vector that takes us from W to X is a. What is the vector that takes us from X to W? Give your answer in the form of an unknown vector.
The vector that we are finding is the opposite to the vector that we have been given. Therefore, we find the vector from X to W by multiplying the vector from W to X by -1. The vector from W to X is a. The calculation is shown below.
The vector that we are finding is the opposite to the vector that we have been given. Therefore, we find the vector from X to W by multiplying the vector from W to X by -1. The vector from W to X is a. The calculation is shown below.
The vector from X to W is -a.
Example 3
Vector b has the column vector form:
Vector b has the column vector form:
What is the column vector for -b?
We find the column vector for -b by multiplying vector b by -1.
We find the column vector for -b by multiplying vector b by -1.
We need to be really carful with the signs because the multiplication of two negative numbers results in a positive. The column vector for -b is shown below:
Scalar
We are able to multiply vectors by a scaler, which means that the size of the vector changes, but the direction of the vector remains the same.
Example 3
Let’s suppose that we have the vector z and the column vector for vector z is:
We are able to multiply vectors by a scaler, which means that the size of the vector changes, but the direction of the vector remains the same.
Example 3
Let’s suppose that we have the vector z and the column vector for vector z is:
What is the vector 3z?
The direction of both of these vectors is the same, but the magnitude (size) is different; 3z had a magnitude that is 3 times greater than the magnitude of z. Therefore, we can obtain the column vector for 3z by multiplying z by 3.
The direction of both of these vectors is the same, but the magnitude (size) is different; 3z had a magnitude that is 3 times greater than the magnitude of z. Therefore, we can obtain the column vector for 3z by multiplying z by 3.
When we multiply a column vector by a scalar value, we multiply both of the components inside the column vector by the scalar value. For this example, we multiply 3 by 4 to obtain the top component in the column vector, and we multiply 3 by -2 to obtain the bottom component in the column vector. The column vector for 3z is shown below.
Scalar values must be positive because if they are negative, they would change the direction of the vector. Scale value can be fractions or decimals and the next example will have a scalar that is a fraction.
Example 4
The vector p has the column vector:
Example 4
The vector p has the column vector:
What is ½ p?
To obtain ½ p, we multiply the p by ½. The column vector for ½ p is shown below.
To obtain ½ p, we multiply the p by ½. The column vector for ½ p is shown below.
Vectors that have been multiplied by a scalar will be parallel to one another.
Scalars are very useful when we are working out the mid-point or ratios of length. The final section in this whole vector section will look at this.
Scalars are very useful when we are working out the mid-point or ratios of length. The final section in this whole vector section will look at this.