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Vector Formulas

Vectors are those which specifies also about the direction along with magnitude and hence differs from scalar. A quantity that has both magnitude and direction. It is typically represented by an arrow whose direction is the same as that of the quantity and whose length is proportional to the quantity's magnitude. On the other hand, a scalar quantity is a quantity that is fully described by its magnitude. The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces that occur in two dimensions. Vector quantities are often represented by scaled vector diagrams. The vector diagram depicts a displacement vector.

There are four operations in vectors -
  • Vector addition,
  • Vector Subtraction,
  • Vector Dot product  and
  • Vector cross product.

Here in this page only Vector addition and subtraction formula have being discussed and  the remaining formulas are discussed in other pages. Vector addition  triangular law and parallelogram law given as below.

Triangular law of addition : If two forces $\vec{A}$and $\vec{B}$ are acting in a same direction then its resultant R will the sum of two vectors.
                                     Vector Addition Formula
                                   Vector Addition

Parallelogram law of addition : If two forces $\vec{A}$ and $\vec{B}$ are represented by the adjacent sides of the parallelogram then their resultant is represented by the diagonal of parallelogram drawn from the same point.
                                      Vector Addition Formula                Parallelogram Formulas

Vector Subtraction : If two forces $\vec{A}$ and $\vec{B}$ are acting in the direction opposite to each other then their resultant R is represented by the difference between the two vectors.
                                      Vector Subtraction Formula
                       Vector Subtraction                                               

Vector Formulas are useful in simple calculations of vectors. It helps to find the resultant vector if the two vectors are given.

Related Calculators
Calculate Vector Adding Vectors Calculator
Angle between Two Vectors Calculator eigen vector calculator
 

Vector Problems

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Below are problems based on vector addition and subtraction which may be helpful for you.

Solved Examples

Question 1: If two forces represented by $\vec{A}$ = 5 $\vec{i}$ + 2 $\vec{j}$ - 3 $\vec{k}$ and $\vec{B}$ = 3 $\vec{i}$ - 2 $\vec{j}$ + 4 $\vec{k}$ are acting in the same direction, calculate the resultant force.
Solution:
 
Let $\vec{A}$ = 5 $\vec{i}$ + 2 $\vec{j}$ - 3 $\vec{k}$,
     $\vec{B}$ = 3 $\vec{i}$ - 2 $\vec{j}$ + 4 $\vec{k}$
$\vec{A}$ + $\vec{B}$ = 5 $\vec{i}$ + 2 $\vec{j}$ - 3 $\vec{k}$ + $\vec{B}$ + 3 $\vec{i}$ - 2 $\vec{j}$ + 4 $\vec{k}$
                                = 8 $\vec{i}$ + $\vec{k}$.
The Resultant force R = 8 $\vec{i}$ + $\vec{k}$.
 

Question 2: If two forces $\vec{A}$ = 2 $\vec{i}$ - 3 $\vec{k}$ and $\vec{B}$ = - 2 $\vec{i}$ + 7 $\vec{j}$ + 4 $\vec{k}$ are acting in the direction opposite to each other. Calculate the Resultant force.
Solution:
 
Let $\vec{A}$ = 2 $\vec{i}$ - 3 $\vec{k}$,
     $\vec{B}$ = - 2 $\vec{i}$ + 7 $\vec{j}$ + 4 $\vec{k}$
$\vec{A}$ - $\vec{B}$ = 2 $\vec{i}$ - 3 $\vec{k}$ - (- 2 $\vec{i}$ + 7 $\vec{j}$ + 4 $\vec{k}$).
                               = 4 $\vec{i}$ - 7 $\vec{j}$ - 7 $\vec{k}$
The Resultant force R = 4 $\vec{i}$ - 7 $\vec{j}$ - 7 $\vec{k}$.
 

More topics in Vector Formulas
Unit Vector Formula Dot Product Formula
Cross Product Formula Vector Projection Formula
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