Question

A vector perpendicular to \[\overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{j}}\,+\overset{\hat{\ }}{\mathop{k}}\,\]
(A) \[\overset{\hat{\ }}{\mathop{i}}\,-\overset{\hat{\ }}{\mathop{j}}\,+\overset{\hat{\ }}{\mathop{k}}\,\]
(B) \[\overset{\hat{\ }}{\mathop{i}}\,-\overset{\hat{\ }}{\mathop{j}}\,-\overset{\hat{\ }}{\mathop{k}}\,\]
(C) \[-\overset{\hat{\ }}{\mathop{i}}\,-\overset{\hat{\ }}{\mathop{j}}\,-\overset{\hat{\ }}{\mathop{k}}\,\]
(D) \[\overset{\hat{\ }}{\mathop{3i}}\,+\overset{\hat{\ }}{\mathop{2j}}\,-\overset{\hat{\ }}{\mathop{5k}}\,\]

Answer 1

Hint: We are given a question asking us to find a vector which is perpendicular to the given vector \[\overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{j}}\,+\overset{\hat{\ }}{\mathop{k}}\,\]. We will let the given vector be \[\overrightarrow{a}=\overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{j}}\,+\overset{\hat{\ }}{\mathop{k}}\,\] and the unknown vector be \[\overrightarrow{b}=\overset{\hat{\ }}{\mathop{xi}}\,+\overset{\hat{\ }}{\mathop{yj}}\,+\overset{\hat{\ }}{\mathop{zk}}\,\]. We are given that the two vectors must be perpendicular to each other. We will use the dot product to find the unknown vector. We know that the dot product of two perpendicular vectors gives the value as 0. So, accordingly we will multiply them and we will get the equation as, \[x+y+z=0\]. Then, we will match the expression with the given options and find the most appropriate option.

Complete step by step answer:
According to the given question, we are given a vector and we are asked to find another vector which is perpendicular to the given vector from the options given to us.
We will name the given vector as,
\[\overrightarrow{a}=\overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{j}}\,+\overset{\hat{\ }}{\mathop{k}}\,\]
And we will suppose the unknown vector be,
\[\overrightarrow{b}=\overset{\hat{\ }}{\mathop{xi}}\,+\overset{\hat{\ }}{\mathop{yj}}\,+\overset{\hat{\ }}{\mathop{zk}}\,\]
Where x, y and z are the components of three coordinate axes.
We are given that the vectors should be perpendicular, that is, the angle between the two vectors should be \[{{90}^{\circ }}\]. We will make use of the dot product of the vectors in order to find the unknown vector.
That is, we have,
\[\overrightarrow{a}.\overrightarrow{b}=0\]
\[\Rightarrow \left( \overset{\hat{\ }}{\mathop{i}}\,+\overset{\hat{\ }}{\mathop{j}}\,+\overset{\hat{\ }}{\mathop{k}}\, \right).\left( \overset{\hat{\ }}{\mathop{xi}}\,+\overset{\hat{\ }}{\mathop{yj}}\,+\overset{\hat{\ }}{\mathop{zk}}\, \right)=0\]
\[\Rightarrow x+y+z=0\]
We obtained the above expression, so in order to get the expression of a perpendicular vector, it should satisfy the above expression.
From the options given, we will take each and check them whether they satisfy the conditions or not. We have,
(A) \[\overset{\hat{\ }}{\mathop{i}}\,-\overset{\hat{\ }}{\mathop{j}}\,+\overset{\hat{\ }}{\mathop{k}}\,\]
We get the value of the expression as,
\[1-1+1\]
\[\Rightarrow 1\ne 0\]
(B) \[\overset{\hat{\ }}{\mathop{i}}\,-\overset{\hat{\ }}{\mathop{j}}\,-\overset{\hat{\ }}{\mathop{k}}\,\]
We get the value of the expression as,
\[1-1-1\]
\[\Rightarrow -1\ne 0\]
(C) \[-\overset{\hat{\ }}{\mathop{i}}\,-\overset{\hat{\ }}{\mathop{j}}\,-\overset{\hat{\ }}{\mathop{k}}\,\]
We get the value of the expression as,
\[-1-1-1\]
\[\Rightarrow -3\ne 0\]
(D) \[\overset{\hat{\ }}{\mathop{3i}}\,+\overset{\hat{\ }}{\mathop{2j}}\,-\overset{\hat{\ }}{\mathop{5k}}\,\]
We get the value of the expression as,
\[3+2-5\]
\[\Rightarrow 0=0\]
Therefore, the perpendicular vector is (D) \[\overset{\hat{\ }}{\mathop{3i}}\,+\overset{\hat{\ }}{\mathop{2j}}\,-\overset{\hat{\ }}{\mathop{5k}}\,\].

So, the correct answer is “Option D”.

Note: The properties of vectors should be known and the various formulae related to it. Also, we can solve the above solution using the formula, \[\overrightarrow{a}.\overrightarrow{b}=|a||b|\cos \theta \], where \[\theta ={{90}^{\circ }}\] and the value of \[\cos {{90}^{\circ }}=0\]. We will substitute these values and we will get the required vector.