Let a,b and c be three non-coplanar vectors and d be a non-zero vector, which is perpendicular to a+b+c. Now, if d=(sinx)(a×b)+(cosy)(b×c)+2(c×a), then minimum value of x2+y2 is equal to

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Let $a, b$ and $c$ be three non-coplanar vectors and $d$ be $a$ non-zero vector, which is perpendicular to $a + b + c$ . Now, if $d = (\sin x) (a \times b) + (\cos y) (b \times c) + 2 (c \times a)$ , then minimum value of $x^{2} + y^{2}$ is equal to

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$\frac{π^{2}}{2}$

$\frac{π^{2}}{4}$

$π^{2}$

$\frac{5 π^{2}}{4}$

answer is D.

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Detailed Solution

$a, b, c$ are non-coplanar $\Rightarrow [a, b, c] \neq 1$ Also, $a \times b, b \times c, c \times a$ are non-coplanar given
$d = \sin x (a \times b) + \cos y (b \times c) + 2 (c \times a)$
Taking dot product with $a + b + c$ , we get
$O = \sin x [a b c] + \cos y [a b c] + 2 [a b c]$
$\Rightarrow \sin x + \cos y + 2 = 0 \Rightarrow \sin x + \cos y = - 2 \Rightarrow x = (4 n - 1) \frac{π}{2}, y = (2 n - 1) π, n \in z$