Question

What is unit vector in the perpendicular to the following vectors $2\hat i + 2\hat j - \hat k$ and $6\hat i - 3\hat j + 2\hat k$
A. $\dfrac{{\hat i + 10\hat j - 18\hat k}}{{5\sqrt {17} }}$
B. $\dfrac{{\hat i - 10\hat j + 18\hat k}}{{5\sqrt {17} }}$
C. $\dfrac{{\hat i - 10\hat j - 18\hat k}}{{5\sqrt {17} }}$
D. $\dfrac{{\hat i + 10\hat j + 18\hat k}}{{5\sqrt {17} }}$

Answer 1

Hint: In order to solve this question we need to understand the vector or cross product of two vectors. So a vector product of two vectors is defined as a vector when cross multiplied with another vector then the resultant vector is in perpendicular direction from both the vectors. So in this question we are going to find the cross product of given two vectors and then later find the unit vector. A unit vector is defined as the vector direction and it is mathematically expressed as the ratio of vectors by its magnitude.

Complete step by step answer:
Consider vector A as, $\vec A = 2\hat i + 2\hat j - \hat k$. And vector B as, $\vec B = 6\hat i - 3\hat j + 2\hat k$. So the cross product of two vectors is defined as, $\vec N = \vec A \times \vec B$. Here, $\vec N$ is the result of cross product of both vectors and is perpendicular to both vectors.

We use the determinant method to find the cross product solution,So putting values to get $\vec N$ as,
\[\;\vec N = (2\hat i + 2\hat j - \hat k) \times (6\hat i - 3\hat j + 2\hat k)\]
\[\Rightarrow \vec N = \left( {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  2&2&{ - 1} \\
  6&{ - 3}&2
\end{array}} \right)\]
\[\Rightarrow \vec N = \hat i[\{ 2 \times 2\} - \{ - 3 \times - 1\} ] - \hat j[\{ 2 \times 2\} - \{ 6 \times - 1\} ] + \hat k[\{ 2 \times - 3\} - \{ 6 \times 2\} ]\]
\[\Rightarrow \vec N = \hat i[4 - 3] - \hat j[4 + 6] + \hat k[ - 6 - 12]\]
\[\Rightarrow \vec N = \hat i - 10\hat j - 18\hat k\]

So magnitude of this vector is given as,
\[\left| {\vec N} \right| = \sqrt {{1^2} + {{10}^2} + {{18}^2}} \]
\[\Rightarrow \left| {\vec N} \right| = \sqrt {425} \]
\[\Rightarrow \left| {\vec N} \right| = \sqrt {25 \times 17} \]
\[\Rightarrow \left| {\vec N} \right| = 5\sqrt {17} \]
So the unit vector in this direction is defined as,
\[\hat N = \dfrac{{\vec N}}{{\left| {\vec N} \right|}}\]
Putting values we get,
\[\therefore \hat N = \dfrac{{\hat i - 10\hat j - 18\hat k}}{{5\sqrt {17} }}\]

Therefore, the correct option is C.

Note: It should be remembered that vectors are those quantities which have both magnitude and direction, also it must follow the triangle rule of vector addition. This law is a compulsory condition for quantity to be considered as a vector, for example current is not a vector as it has both magnitude and direction but it does not follow the triangle rule of vector addition, because of Kirchhoff’s junction rule.