Question

If the vectors \[\bar{i}-\bar{j},\bar{j}+\bar{k}\] and \[\bar{a}\] from a triangle, then \[\bar{a}\] may be
(a) \[-\bar{i}-\bar{k}\]
(b) \[\bar{i}-2\bar{j}-\bar{k}\]
(c) \[2\bar{i}+\bar{j}+\bar{k}\]
(d) \[\bar{i}+\bar{k}\]

Answer 1

Hint: We solve this problem by considering all the possibilities of the triangle.
We use the condition that if a triangle is formed by three vectors \[\bar{x},\bar{y},\bar{z}\] then
\[\bar{x}+\bar{y}+\bar{z}=0\]
Then we assume that any side of the triangle can be reversed so that the vector will get the negative sign and again we use the same above condition to get the required vector.

Complete step by step answer:
We are given with the three vectors as \[\bar{i}-\bar{j},\bar{j}+\bar{k}\] and \[\bar{a}\] from a triangle
Let us assume that the three vectors as
\[\begin{align}
  & \Rightarrow \bar{x}=\bar{i}-\bar{j} \\
 & \Rightarrow \bar{y}=\bar{j}+\bar{k} \\
 & \Rightarrow \bar{z}=\bar{a} \\
\end{align}\]
Let us take a general figure of the triangle formed by three vectors as follows

We know that the condition that if a triangle is formed by three vectors \[\bar{x},\bar{y},\bar{z}\] then
\[\bar{x}+\bar{y}+\bar{z}=0\]
By using the above condition to given vectors then we get
\[\begin{align}
  & \Rightarrow \bar{i}-\bar{j}+\bar{j}+\bar{k}+\bar{a}=0 \\
& \Rightarrow \bar{a}=-\bar{i}-\bar{k} \\
\end{align}\]
Here, we can see that the vector \[\bar{a}\] can be \[-\bar{i}-\bar{k}\]
We know that there is a possibility of any one side of the triangle is reversed.
Let us assume that \[\bar{x}\] is reversed then we get the figure as

Here, we can change the direction of vector \[\bar{x}\] so that it will become negative as follows

By using the triangle condition then we get
\[\begin{align}
  & \Rightarrow -\left( \bar{i}-\bar{j} \right)+\left( \bar{j}+\bar{k} \right)+\bar{a}=0 \\
 & \Rightarrow \bar{a}=\bar{i}-2\bar{j}-\bar{k} \\
\end{align}\]
Here, we can see that the vector \[\bar{a}\] can be \[\bar{i}-2\bar{j}-\bar{k}\]
We know that there is a possibility of any one side of the triangle is reversed.
Let us assume that \[\bar{y}\] is reversed then we get the figure as

Here, we can change the direction of vector \[\bar{y}\] so that it will become negative as follows

By using the triangle condition then we get
\[\begin{align}
  & \Rightarrow \left( \bar{i}-\bar{j} \right)-\left( \bar{j}+\bar{k} \right)+\bar{a}=0 \\
 & \Rightarrow \bar{a}=-\bar{i}+2\bar{j}+\bar{k} \\
\end{align}\]
Here, we can see that the vector \[\bar{a}\] can be \[-\bar{i}+2\bar{j}+\bar{k}\]
We know that there is a possibility of any one side of the triangle is reversed.
Let us assume that \[\bar{z}\] is reversed then we get the figure as

Here, we can change the direction of vector \[\bar{z}\] so that it will become negative as follows

By using the triangle condition then we get
\[\begin{align}
  & \Rightarrow \left( \bar{i}-\bar{j} \right)+\left( \bar{j}+\bar{k} \right)-\bar{a}=0 \\
 & \Rightarrow \bar{a}=\bar{i}+\bar{k} \\
\end{align}\]
Here, we can see that the vector \[\bar{a}\] can be \[\bar{i}+\bar{k}\]
Therefore we can conclude that the possible values of vector \[\bar{a}\] are \[\left( -\bar{i}-\bar{k} \right),\left( \bar{i}-2\bar{j}-\bar{k} \right),\left( -\bar{i}+2\bar{j}+\bar{k} \right),\left( \bar{i}+\bar{k} \right)\]
So, option (a), option (b) and option (d) are correct answers.

Note:
Students may do mistake without taking the possibilities of reversing the direction of each side of the triangle.
Initially we have the triangle as

By using the triangle condition to given vectors then we get
\[\begin{align}
  & \Rightarrow \bar{i}-\bar{j}+\bar{j}+\bar{k}+\bar{a}=0 \\
 & \Rightarrow \bar{a}=-\bar{i}-\bar{k} \\
\end{align}\]
Here, we can see that the vector \[\bar{a}\] can be \[-\bar{i}-\bar{k}\]
Here, we have other three possibilities of reversing the directions of \[\bar{x},\bar{y},\bar{z}\]
So we get three more possibilities for the value of \[\bar{a}\] but students may take only first possibility and conclude the answer.