Question

With respect to a rectangular cartesian coordinate system, three vectors are expressed as $\vec a = 4\hat i - \hat j,\vec b = - 3\hat i + 2\hat j$ and $\vec c = - \hat k$ where $\hat i,\hat j,\hat k$ are unit vectors, along X, Y and Z axis respectively. The unit vector $\hat r$ along the direction of sum of these vectors is
A. $\hat r = \dfrac{1}{{\sqrt 3 }}(\hat i + \hat j - \hat k)$
B. $\hat r = \dfrac{1}{{\sqrt 2 }}(\hat i + \hat j - \hat k)$
C. $\hat r = \dfrac{1}{{\sqrt 3 }}(\hat i - \hat j + \hat k)$
D. $\hat r = \dfrac{1}{{\sqrt 2 }}(\hat i + \hat j + \hat k)$

Answer 1

Hint: Firstly, we will find $\vec r$ as it is given that $\hat r$ is the sum of three vectors using the fact that $\vec r = \vec a + \vec b + \vec c$. Lastly, we have to find the unit vector $\hat r$ along the direction of sum of these vectors using the formula $\hat r = \dfrac{{\vec r}}{{|\vec r|}}$ where $|\vec r| = \sqrt {{x^2} + {y^2} + {z^2}} $ then after simplifying we will get the required solution.

Formula Used:
$\hat r = \dfrac{{\vec r}}{{|\vec r|}}$
$|\vec r| = \sqrt {{x^2} + {y^2} + {z^2}} $

Complete step by step solution:
We have given three vectors.
$\vec a = 4\hat i - \hat j$
$\vec b = - 3\hat i + 2\hat j$
$\,\vec c = - \hat k$
$\vec r = \vec a + \vec b + \vec c$
$\vec r = 4\hat i - j - 3\hat i + 2\hat j - \hat k$
$\vec r = \hat i + \hat j - \hat k$
The unit vector is
$\hat r = \dfrac{{\vec r}}{{|\vec r|}}$
$\hat r = \dfrac{{\hat i + \hat j - \hat k}}{{\sqrt {1 + 1 + 1} }}$
$\hat r = \dfrac{{\hat i + \hat j - \hat k}}{{\sqrt 3 }}$
$\hat r = \dfrac{1}{{\sqrt 3 }}(\hat i + \hat j - \hat k)$

Option ‘A’ is correct

Additional Information: Vectors area unit quantities consisting of magnitude and direction. The scale of the vector is set by its magnitude. It’s diagrammatic by a line with an Associate in Nursing arrow, with the length of the road representing the vector’s magnitude and therefore the arrow indicating the direction. Force and speed area unit the 2 samples of vectors.
When the magnitude and direction of 2 vectors are identical, they're equal. Vector is additionally called a geometrician vector or spatial vector. Vectors represent physical quantities like displacement, velocity, and acceleration.

Note: When you get to solve such problems you need to know that in a rectangle only three coordinates are sufficient to get the value of all the terms.