Question

If \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\] are three vectors such that \[\vec{a}.\vec{b}=\vec{a}.\vec{c}\] then show that \[\vec{a}=0\] or \[\vec{b}=\vec{c}\]or \[\vec{a}\] is perpendicular to \[\vec{b}-\vec{c}\].

Answer 1

Hint: In order to prove the given question that is if \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\] are three vectors such that \[\vec{a}.\vec{b}=\vec{a}.\vec{c}\] then show that \[\vec{a}=0\] or \[\vec{b}=\vec{c}\]or \[\vec{a}\] is perpendicular to \[\vec{b}-\vec{c}\]. Operate the given relation between the given three vectors \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\] that is \[\vec{a}.\vec{b}=\vec{a}.\vec{c}\]. Take all the terms to the left-hand side then take the vector \[\vec{a}\] in common. After this apply one of the results of vectors that is if \[\vec{a}\left( \vec{b}-\vec{c} \right)=0\] then it means that either \[\vec{b}=\vec{c}\] and by Orthogonal Vector Theorem which states that two vectors \[\vec{A}\] and \[\vec{B}\] are perpendicular if and only if their dot product is zero.

Complete step by step solution:
According to the question, given relation between the three vectors \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\] is as follows:
\[\Rightarrow \vec{a}.\vec{b}=\vec{a}.\vec{c}\]
We have to show that \[\vec{a}=0\] or \[\vec{b}=\vec{c}\] or \[\vec{a}\] is perpendicular to \[\vec{b}-\vec{c}\].
Proof: In order to prove the above statement, Operate the given relation between the given three vectors \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\] that is \[\vec{a}.\vec{b}=\vec{a}.\vec{c}\]. Take all the terms to the left-hand side, we get:
\[\Rightarrow \vec{a}.\vec{b}-\vec{a}.\vec{c}=0\]
As we can see that the vector \[\vec{a}\] is there in both the terms in the left-hand side of the equation, apply the distributive property of the dot product that is if there are three vectors \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\]then \[\vec{a}.\vec{b}+\vec{a}.\vec{c}=\vec{a}.\left( \vec{b}+\vec{c} \right)\] in the above equation, we get:
\[\Rightarrow \vec{a}.\left( \vec{b}-\vec{c} \right)=0\]
This implies that \[\vec{a}=0\] or \[\vec{b}=\vec{c}\].
Also, according to the Orthogonal Vector Theorem which states that two vectors \[\vec{A}\] and \[\vec{B}\] are perpendicular if and only if their dot product is zero that is \[\vec{a}\] is perpendicular to \[\vec{b}-\vec{c}\].
Therefore, we have proved that if \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\] are three vectors such that \[\vec{a}.\vec{b}=\vec{a}.\vec{c}\]then either\[\vec{a}=0\] or \[\vec{b}=\vec{c}\]or \[\vec{a}\] is perpendicular to \[\vec{b}-\vec{c}\].

Note: Students get confused with the concept of orthogonal vector theorem. It is important to remember that according to the Orthogonal Vector Theorem if two vectors \[\vec{A}\] and \[\vec{B}\] are perpendicular if and only if their dot product is zero.