JEE MAIN - Mathematics (2021 - 25th July Evening Shift - No. 8)
Let a, b and c be distinct positive numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i+\widehat k$$ and $$c\widehat i + c\widehat j + b\widehat k$$ are co-planar, then c is equal to :
$${2 \over {{1 \over a} + {1 \over b}}}$$
$${{a + b} \over 2}$$
$${1 \over a} + {1 \over b}$$
$$\sqrt {ab} $$
Explanation
Because vectors are coplanar
Hence, $$\left| {\matrix{ a & a & c \cr 1 & 0 & 1 \cr c & c & b \cr } } \right| = 0$$
$$ \Rightarrow {c^2} = ab \Rightarrow c = \sqrt {ab} $$
Hence, $$\left| {\matrix{ a & a & c \cr 1 & 0 & 1 \cr c & c & b \cr } } \right| = 0$$
$$ \Rightarrow {c^2} = ab \Rightarrow c = \sqrt {ab} $$
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