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If $a = i + j + k, b = 2 i + 3 j, c = 3 i + 5 j - 2 k,$ and $d = k - j,$ then the ratio of modules of $b - a a n d d - c i s$

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1:3

3:1

2:3

4:1

answer is B.

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Detailed Solution

$b - a = i + 2 j - k \Rightarrow |b - a| = \sqrt{1 + 4 + 1} = \sqrt{6} d - c = - 3 i - 6 j + 3 k \Rightarrow |d - c| = \sqrt{9 + 36 + 9} = \sqrt{54} \frac{|b - a|}{|d - c|} = \sqrt{\frac{6}{54}} = \frac{1}{3}$

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