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3-1.Vectors
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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
A$\hat r = \frac{1}{{\sqrt 3 }}(\hat i + \hat j - \hat k)$
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$\hat r = \frac{1}{{\sqrt 2 }}(\hat i + \hat j + \hat k)$
Solution
(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$
$\hat r = \frac{{\vec r}}{{|r|}} = \frac{{\hat i + \hat j – \hat k}}{{\sqrt {{1^2} + {1^2} + {{( – 1)}^2}} }} = \frac{{\hat i + \hat j – \hat k}}{{\sqrt 3 }}$
$ = 4\hat i – \hat j – 3\hat i + 2\hat j – \hat k$
$ = \hat i + \hat j – \hat k$
$\hat r = \frac{{\vec r}}{{|r|}} = \frac{{\hat i + \hat j – \hat k}}{{\sqrt {{1^2} + {1^2} + {{( – 1)}^2}} }} = \frac{{\hat i + \hat j – \hat k}}{{\sqrt 3 }}$
Standard 11
Physics
