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Question: Which of the following is not a unit vector? A. \(\dfrac{{2\hat i + 3\hat j + 2\hat k}}{{\sqrt {17...

Which of the following is not a unit vector?
A. 2i^+3j^+2k^17\dfrac{{2\hat i + 3\hat j + 2\hat k}}{{\sqrt {17} }}
B. 2i^−3j^−k^17\dfrac{{2\hat i - 3\hat j - \hat k}}{{\sqrt {17} }}
C. i^−j^−k^3\dfrac{{\hat i - \hat j - \hat k}}{{\sqrt 3 }}
D. i^+k^3\dfrac{{\hat i + \hat k}}{{\sqrt 3 }}

Explanation

Solution

We know that a unit vector is a vector whose magnitude is equal to 11 . Here, we will find the magnitude of each vector from the given option and whose magnitude will be unity that will be a unit vector.

Complete step by step answer:
For a given vector in the form of ai^+bj^+ck^a\hat i + b\hat j + c\hat k the magnitude of such vectors is given bya2+b2+c2\sqrt {{a^2} + {b^2} + {c^2}} . i^\hat i, j^\hat j And k^\hat k are the unit vectors of length one in the direction of XX axis, YY axis and ZZ axis. And aa bb cc are the magnitudes of a vector which makes intercept on these axes respectively.
Firstly, we will find the magnitude of given vector 2i^+3j^+2k^17\dfrac{{2\hat i + 3\hat j + 2\hat k}}{{\sqrt {17} }}
Magnitude =4+9+417=1 = \dfrac{{\sqrt {4 + 9 + 4} }}{{\sqrt {17} }}= 1 Unit.
Hence a given vector in option A is a unit vector.
Now, magnitude of vector 2i^−3j^−k^17\dfrac{{2\hat i - 3\hat j - \hat k}}{{\sqrt {17} }} is given by
⇒1+1+13=1\Rightarrow \dfrac{{\sqrt {1 + 1 + 1} }}{{\sqrt 3 }}= 1 Unit.
Hence a given vector in option B is a unit vector.
Now, magnitude of vector i^−j^−k^3\dfrac{{\hat i - \hat j - \hat k}}{{\sqrt 3 }} is given by
⇒1+1+13=1\Rightarrow \dfrac{{\sqrt {1 + 1 + 1} }}{{\sqrt 3 }}=1 Unit.
Hence a given vector in option C is also a unit vector.
Now, magnitude of vector i^+k^3\dfrac{{\hat i + \hat k}}{{\sqrt 3 }} is given by
⇒1+13=23\Rightarrow \dfrac{{\sqrt {1 + 1} }}{{\sqrt 3 }}=\sqrt {\dfrac{2}{3}}
Which does not equal unity. So, the vector given in option D is not a unit vector.

Hence, option D is the answer.

Note: The only purpose of a unit vector is to give the idea of direction in which a vector is headed. A unit vector is also known as a Direction vector. There is a radial unit vector which lies along the surface and there is a normal unit vector which is perpendicular to the surface.