Question

Given two vectors \[{\text{A}} = 4{\text{i}} + 3{\text{j}}\;{\text{and}}\;{\text{B}} = 5{\text{i}} - 2{\text{j}}\] how do you write an expression for the vector difference \[{\text{A}} - {\text{B}}\] using unit vectors?

Answer 1

Hint: Consider the two vectors “a” and “b”. If vector “a” is to be subtracted from vector “b”, we must find the negative of the negative of vector “a” and then it should be added to “b” in order to find the subtraction of two vectors according to triangle law of vectors.
It can be mathematically understood as follows: $b - a = b + ( - a)$.

Formula used: Vector subtraction of two vectors “A” and “B” which is given as follows
\[{\text{A}} - {\text{B}} = {\text{A}} + \left( { - {\text{B}}} \right)\]

Complete step-by-step solution:
 In order to find the vector difference \[{\text{A}} - {\text{B}}\]of two given vectors \[{\text{A}} = 4{\text{i}} + 3{\text{j}}\;{\text{and}}\;{\text{B}} = 5{\text{i}} - 2{\text{j}}\], we have to first find the negative of the vector which is going to be subtracted from the another vector.
So in the given vector difference expression \[{\text{A}} - {\text{B}}\], we can see that vector “B” is being subtracted from vector “A”, therefore we will find the negative of the vector “B” as follows
\[ - {\text{B}} = - \left( {5{\text{i}} - 2{\text{j}}} \right) = \left( { - 5{\text{i}} + 2{\text{j}}} \right)\]
Now we will add the negative of vector “B” to vector “A” in order to find the vector difference of the expression \[{\text{A}} - {\text{B}}\]
\[
  {\text{A}} + \left( { - {\text{B}}} \right) = \left( {4{\text{i}} + 3{\text{j}}} \right) + \left( { - 5{\text{i}} + 2{\text{j}}} \right) \\
   = 4{\text{i}} + 3{\text{j}} - 5{\text{i}} + 2{\text{j}} \\
   = - {\text{i}} + 5{\text{j}} \\
 \]
Therefore required vector difference \[{\text{A}} - {\text{B}}\] is equals to \[\left( { - {\text{i}} + 5{\text{j}}} \right)\]

Note: In order to understand that why we have find the negative term of the vector which is to be subtracted and then added it to the another vector to eventually find the subtraction of the two vectors, you should refer to the law of parallelogram vectors or the triangle law of vectors (which is itself a part of the parallelogram law).