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Question: If‌ ‌vector‌ ‌\(\overrightarrow{AB}=2\hat{i}-\hat{j}+\hat{k}\)‌ ‌and‌ ‌\(\overrightarrow{OB}=3\hat{i...

If‌ ‌vector‌ ‌AB→=2i^−j^+k^\overrightarrow{AB}=2\hat{i}-\hat{j}+\hat{k}‌ ‌and‌ ‌OB→=3i^−4j^+4k^\overrightarrow{OB}=3\hat{i}-4\hat{j}+4\hat{k}‌. Find‌ ‌the‌ ‌position‌ ‌vector‌ ‌OA→\overrightarrow{OA}‌.

Explanation

Solution

In this question we have been asked to find the position vector of OA→\overrightarrow{OA} using the given information stated as the position vectors of AB→=2i^−j^+k^\overrightarrow{AB}=2\hat{i}-\hat{j}+\hat{k} and OB→=3i^−4j^+4k^\overrightarrow{OB}=3\hat{i}-4\hat{j}+4\hat{k}. From the basic concepts we know that the position vector of AB→\overrightarrow{AB} is generally given as OB→−OA→\overrightarrow{OB}-\overrightarrow{OA} .

Complete step by step solution:
Now considering from the question we have been asked to find the position vector of OA→\overrightarrow{OA} using the given information stated as the position vectors of AB→=2i^−j^+k^\overrightarrow{AB}=2\hat{i}-\hat{j}+\hat{k} and OB→=3i^−4j^+4k^\overrightarrow{OB}=3\hat{i}-4\hat{j}+4\hat{k}.
From the basic concepts of vectors we know that the position vector of AB→\overrightarrow{AB} is generally given as OB→−OA→\overrightarrow{OB}-\overrightarrow{OA} .
Now by substituting the given vectors position vectors in this expression we will have ⇒2i^−j^+k^=3i^−4j^+4k^−OA→\Rightarrow 2\hat{i}-\hat{j}+\hat{k}=3\hat{i}-4\hat{j}+4\hat{k}-\overrightarrow{OA}.
By transforming the required position vector to the left side and the other terms to the right side we will have
⇒OA→+2i^−j^+k^=3i^−4j^+4k^ ⇒OA→=3i^−4j^+4k^−2i^+j^−k^ \begin{aligned} & \Rightarrow \overrightarrow{OA}+2\hat{i}-\hat{j}+\hat{k}=3\hat{i}-4\hat{j}+4\hat{k} \\\ & \Rightarrow \overrightarrow{OA}=3\hat{i}-4\hat{j}+4\hat{k}-2\hat{i}+\hat{j}-\hat{k} \\\ \end{aligned}
By further simplifying this expression by performing addition and subtraction of similar terms as per the requirement we will have ⇒OA→=i^−3j^+3k^\Rightarrow \overrightarrow{OA}=\hat{i}-3\hat{j}+3\hat{k}
Therefore we can conclude that when the position vectors of AB→=2i^−j^+k^\overrightarrow{AB}=2\hat{i}-\hat{j}+\hat{k} and OB→=3i^−4j^+4k^\overrightarrow{OB}=3\hat{i}-4\hat{j}+4\hat{k} then the position vector of OA→\overrightarrow{OA} is given as OA→=i^−3j^+3k^\overrightarrow{OA}=\hat{i}-3\hat{j}+3\hat{k} .

Note: During the process of answering questions of this type we should be sure with our concepts that we are going to apply and the calculations that we are going to perform in between. This is a very simple question and can be answered in a short span of time. Generally some people confuse and consider AB→\overrightarrow{AB} as OA→−OB→\overrightarrow{OA}-\overrightarrow{OB} which gives the result as
⇒2i^−j^+k^=OA→−(3i^−4j^+4k^) ⇒OA→=2i^−j^+k^+3i^−4j^+4k^ ⇒OA→=5i^−5j^+5k^ \begin{aligned} & \Rightarrow 2\hat{i}-\hat{j}+\hat{k}=\overrightarrow{OA}-\left( 3\hat{i}-4\hat{j}+4\hat{k} \right) \\\ & \Rightarrow \overrightarrow{OA}=2\hat{i}-\hat{j}+\hat{k}+3\hat{i}-4\hat{j}+4\hat{k} \\\ & \Rightarrow \overrightarrow{OA}=5\hat{i}-5\hat{j}+5\hat{k} \\\ \end{aligned}
which is a wrong answer.