Basic Operations with Vectors
Vector (Euclidian Vector) is a geometric object that has magnitude (length) and direction. The vector operations are: Multiplication and Division of a Vector with Scalar,Vector Addition, and Vector Subtraction.
Multiplication and Division of a Vector with Scalar - If vector is multiplied by a positive scalar the magnitude of a vector will be increase by that amount. If vector is multiplied by a negative scalar than the direction will be changed. If vector is divided by a scalar value than the magnitude of the vector will be decereased by that ammount.
Figure 2.1 - Examples of mutliplication and division of a vector by scalar
As seen from Figure 2.1 the vector \(\vec{F}\) is multiplied with positive scalar (number 2) the magnitude of the origianl vector \(\vec{F}\) was increased two times i.e. the new vector (\(2\vec{F}\)) is two times larger than the original vector \(\vec{F}.\) If the original vector \(\vec{F}\) is multiplied with the negative scalar (number -1) then the direction of the vector \(\vec{F}\) is changed. The new vector \(-\vec{F}\) has opposite direction when compared to the origianl vector \(\vec{F}.\) If the vector is divided by a negative scalar value for example \(-\frac{1}{2}\) then the magnitude and direction will change. The new vector \(-0.5\vec{F}\) will have opposite direction than vector \(\vec{F}\) and it will be two times smaller than original vector since it was divided with negative number 2.
Vector Addition - Two vectors are added together using parallelogram law of addition (account their magnitudes and directions). Example of parallelogram law of addition is shown in Figure 2.2.
Figure 2.2 - The example procedure of parallelogram law
In Figure 2.2 the resultant vector \(\vec{R}\) is calculated as addition of vectors \(\vec{F}_1\) and \(\vec{F}_2\) which can be written as: \begin{eqnarray} \vec{R} &=& \vec{F}_1 + \vec{F}_2. \end{eqnarray} There is also a special case of the parallegram law which can be used for vector addition called triangle rule. In this case the vactor \(\vec{F}_2\) is added to vector \(\vec{F}_1\) by connecting the head of vector \(\vec{F}_1\) to the tail of \(\vec{F}_2\). The resultant vector \(\vec{R}\) starts from the tail of vector \(\vec{F}_1\) and end at the head of vector \(\vec{F}_2\). The example of traingle rule is shown in Figure 2.3.
Figure 2.3 - The example procedure of triangle rule
The special case of vector addition of two vectors \(\vec{F}_1\) and \(\vec{F}_2\) is that these two are colinear. If they are colinear, i.e. have the same line of action than the classic parallelogram rule reduces to scalar addition of vectors \(\vec{F}_1\) and \(\vec{F}_2\) (\(\vec{R} = \vec{F}_1 + \vec{F}_2\)).
Vector Subtraction - the difference between two vectors \(\vec{F}_1\) nad \(\vec{F}_2\) can be written in the following form $$ \vec{R} = \vec{F}_1 - \vec{F}_2 = \vec{F}_1 + (-\vec{F}_2)$$ The graphical procedure of vector subtraction using parallelogram or triangle rule is shown in Figure 2.4.
Figure 2.4 - Subtraction of vectors \(\vec{F}_1\) and \(\vec{F}_2\).
Addition of Several Forces - If three or more forces have to be added together to determine the resultant force the successive application of the parallelogram law should be carred out. Example of addition of 5 forces together to determine the resultant force is shown in Figure 2.5.
Figure 2.5 - Adition of 5 forces
Multiplication and Division of a Vector with Scalar - If vector is multiplied by a positive scalar the magnitude of a vector will be increase by that amount. If vector is multiplied by a negative scalar than the direction will be changed. If vector is divided by a scalar value than the magnitude of the vector will be decereased by that ammount.
As seen from Figure 2.1 the vector \(\vec{F}\) is multiplied with positive scalar (number 2) the magnitude of the origianl vector \(\vec{F}\) was increased two times i.e. the new vector (\(2\vec{F}\)) is two times larger than the original vector \(\vec{F}.\) If the original vector \(\vec{F}\) is multiplied with the negative scalar (number -1) then the direction of the vector \(\vec{F}\) is changed. The new vector \(-\vec{F}\) has opposite direction when compared to the origianl vector \(\vec{F}.\) If the vector is divided by a negative scalar value for example \(-\frac{1}{2}\) then the magnitude and direction will change. The new vector \(-0.5\vec{F}\) will have opposite direction than vector \(\vec{F}\) and it will be two times smaller than original vector since it was divided with negative number 2.
Vector Addition - Two vectors are added together using parallelogram law of addition (account their magnitudes and directions). Example of parallelogram law of addition is shown in Figure 2.2.
In Figure 2.2 the resultant vector \(\vec{R}\) is calculated as addition of vectors \(\vec{F}_1\) and \(\vec{F}_2\) which can be written as: \begin{eqnarray} \vec{R} &=& \vec{F}_1 + \vec{F}_2. \end{eqnarray} There is also a special case of the parallegram law which can be used for vector addition called triangle rule. In this case the vactor \(\vec{F}_2\) is added to vector \(\vec{F}_1\) by connecting the head of vector \(\vec{F}_1\) to the tail of \(\vec{F}_2\). The resultant vector \(\vec{R}\) starts from the tail of vector \(\vec{F}_1\) and end at the head of vector \(\vec{F}_2\). The example of traingle rule is shown in Figure 2.3.
The special case of vector addition of two vectors \(\vec{F}_1\) and \(\vec{F}_2\) is that these two are colinear. If they are colinear, i.e. have the same line of action than the classic parallelogram rule reduces to scalar addition of vectors \(\vec{F}_1\) and \(\vec{F}_2\) (\(\vec{R} = \vec{F}_1 + \vec{F}_2\)).
Vector Subtraction - the difference between two vectors \(\vec{F}_1\) nad \(\vec{F}_2\) can be written in the following form $$ \vec{R} = \vec{F}_1 - \vec{F}_2 = \vec{F}_1 + (-\vec{F}_2)$$ The graphical procedure of vector subtraction using parallelogram or triangle rule is shown in Figure 2.4.
Addition of Several Forces - If three or more forces have to be added together to determine the resultant force the successive application of the parallelogram law should be carred out. Example of addition of 5 forces together to determine the resultant force is shown in Figure 2.5.