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Two vector $2 \hat{\mathbf{i}}+m \hat{\mathbf{j}}-3 n \hat{\mathbf{k}}$ and $5 \hat{\mathbf{i}}+3 m \hat{\mathbf{j}}+n \hat{\mathbf{k}}$ are such
that their magnitudes are respectively $\sqrt{14}$ and $\sqrt{35}$, where
$\mathrm{m}, \mathrm{n}$ are integers. Which one of the following is correct?
Options:
that their magnitudes are respectively $\sqrt{14}$ and $\sqrt{35}$, where
$\mathrm{m}, \mathrm{n}$ are integers. Which one of the following is correct?
Solution:
2109 Upvotes
Verified Answer
The correct answer is:
$\mathrm{m}$ takes 2 value, $\mathrm{n}$ takes 2 values
Vectors are $2 \mathrm{i}+\mathrm{mj}-3 \mathrm{nk}$ and $5 \mathrm{i}+3 \mathrm{mj}+\mathrm{nk}$
$|2 \mathrm{i}+\mathrm{mj}-3 \mathrm{nk}|=\sqrt{14}$
and $|5 \mathrm{i}+3 \mathrm{mj}+\mathrm{nk}|=\sqrt{35}$
$\sqrt{2^{2}+m^{2}+(-3 n)^{2}}=\sqrt{14}$
or, $4+\mathrm{m}^{2}+9 \mathrm{n}^{2}=14$
or, $\mathrm{m}^{2}+9 \mathrm{n}^{2}=10$
From (ii)
$\sqrt{5^{2}+(3 m)^{2}+n^{2}}=\sqrt{35}$
or, $25+9 \mathrm{~m}^{2}+\mathrm{n}^{2}=35$
or, $9 \mathrm{~m}^{2}+\mathrm{n}^{2}=10$
From (iii) and (iv) $m^{2}+9 n^{2}=9 m^{2}+n^{2}$
or, $8 \mathrm{n}^{2}=8 \mathrm{~m}^{2}$
or, $\mathrm{n}^{2}=\mathrm{m}^{2}$
$\mathrm{n}=\pm \mathrm{m}$
n takes 2 values and $\mathrm{m}$ takes 2 values.
$|2 \mathrm{i}+\mathrm{mj}-3 \mathrm{nk}|=\sqrt{14}$
and $|5 \mathrm{i}+3 \mathrm{mj}+\mathrm{nk}|=\sqrt{35}$
$\sqrt{2^{2}+m^{2}+(-3 n)^{2}}=\sqrt{14}$
or, $4+\mathrm{m}^{2}+9 \mathrm{n}^{2}=14$
or, $\mathrm{m}^{2}+9 \mathrm{n}^{2}=10$
From (ii)
$\sqrt{5^{2}+(3 m)^{2}+n^{2}}=\sqrt{35}$
or, $25+9 \mathrm{~m}^{2}+\mathrm{n}^{2}=35$
or, $9 \mathrm{~m}^{2}+\mathrm{n}^{2}=10$
From (iii) and (iv) $m^{2}+9 n^{2}=9 m^{2}+n^{2}$
or, $8 \mathrm{n}^{2}=8 \mathrm{~m}^{2}$
or, $\mathrm{n}^{2}=\mathrm{m}^{2}$
$\mathrm{n}=\pm \mathrm{m}$
n takes 2 values and $\mathrm{m}$ takes 2 values.
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