With respect to a rectangular cartesian coordinate system, three … - Vidyadip

MCQ

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 the $X, Y $ and $Z-$axis respectively. The unit vectors $\hat r$ along the direction of sum of these vector is

✓
$\hat r = \frac{1}{{\sqrt 3 }}(\hat i + \hat j - \hat k)$
B
$\hat r = \frac{1}{{\sqrt 2 }}(\hat i + \hat j - \hat k)$
C
$\hat r = \frac{1}{3}(\hat i - \hat j + \hat k)$
D
$\hat r = \frac{1}{{\sqrt 2 }}(\hat i + \hat j + \hat k)$

✓

Answer

Correct option: A.

$\hat r = \frac{1}{{\sqrt 3 }}(\hat i + \hat j - \hat k)$

a
(a) $\vec r = \vec a + \vec b + \vec c$

$ = 4\hat i - \hat j - 3\hat i + 2\hat j - \hat k$

$ = \hat i + \hat j - \hat k$

$\hat r = \frac{{\vec r}}{{|r|}} = \frac{{\hat i + \hat j - \hat k}}{{\sqrt {{1^2} + {1^2} + {{( - 1)}^2}} }} = \frac{{\hat i + \hat j - \hat k}}{{\sqrt 3 }}$

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