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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 \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