given that a×b=3i + j +k a×c=−i −2j +k find i)c×a ii)a×(b×c) iii)(a×b)•(a×c)


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Question Number 43856 by MASANJA J last updated on 16/Sep/18

$g i v e n t h a t$ $a \times b = 3 i + j + k$ $a \times c = - i - 2 j + k$ $f i n d$ $i) c \times a$ $i i) a \times (b \times c)$ $i i i) (a \times b) ∙ (a \times c)$
	Answered by tanmay.chaudhury50@gmail.com last updated on 16/Sep/18
	$i)c×a=−a×c c×a=−(−i−2j+k)=i+2j−k ii)a×(b×c)=(a.c)b−(a.b)c so a×(b×c)=(accosθ_1 )−(abcosθ_2 ) now a×c=acsinθ_1 n^→ ∣a×c∣=acsinθ_1 sinθ_1 =((√((−1)^2 +(−2)^2 +(1)^2 ))/(ac))=(((√6) )/(ac)) cosθ_1 =((√(a^2 c^2 −6))/(ac)) sinθ_2 =((√(3^2 +1^2 +1^2 ))/(ab))=((√(11))/(ab)) cosθ_2 =((√(a^2 b^2 −11))/(ab)) a×(b×c)=(a.c)b−(a.b)c =(accosθ_1 )b−(abcosθ_2 )c a×(b×c)={(√(a^2 c^2 −6)) }b^→ −{(√(a^2 b^2 −11)) }c^→$
	$i) c \times a = - a \times c$ $c \times a = - (- i - 2 j + k) = i + 2 j - k$ $i i) a \times (b \times c) = (a . c) b - (a . b) c$ $s o a \times (b \times c) = (a c c o s θ_{1}) - (a b c o s θ_{2})$ $n o w$ $a \times c = a c s i n θ_{1} \vec{n}$ $∣ a \times c ∣= a c s i n θ_{1}$ $s i n θ_{1} = \frac{\sqrt{{(- 1)}^{2} + {(- 2)}^{2} + {(1)}^{2}}}{a c} = \frac{\sqrt{6}}{a c}$ $c o s θ_{1} = \frac{\sqrt{a^{2} c^{2} - 6}}{a c}$ $s i n θ_{2} = \frac{\sqrt{3^{2} + 1^{2} + 1^{2}}}{a b} = \frac{\sqrt{11}}{a b}$ $c o s θ_{2} = \frac{\sqrt{a^{2} b^{2} - 11}}{a b}$ $a \times (b \times c) = (a . c) b - (a . b) c$ $= (a c c o s θ_{1}) b - (a b c o s θ_{2}) c$ $a \times (b \times c) = {\sqrt{a^{2} c^{2} - 6}} \vec{b} - {\sqrt{a^{2} b^{2} - 11}} \vec{c}$
	Answered by tanmay.chaudhury50@gmail.com last updated on 16/Sep/18

	$i i i) (a \times b) . (a \times c) = {(3 \times - 1) + (1 \times - 2) + (1 \times 1)$ $= {- 3 - 2 + 1} = - 4$
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