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Question

(B) We are given the expression $(a + b) \cdot ((b + c) 	imes (a + b + c))$.First,simplify the cross product term: $(b + c) 	imes (a + b + c) = b

Vedclass · Accepted Answer

$ [a b c] $

Vedclass · Answer

(B) We are given the expression $(a + b) \cdot ((b + c) 	imes (a + b + c))$.First,simplify the cross product term: $(b + c) 	imes (a + b + c) = b 	imes a + b 	imes b + b 	imes c + c 	imes a + c 	imes b + c 	imes c$.Since $b 	imes b = 0$ and $c 	imes c = 0$,and using the property $c 	imes b = -(b 	imes c)$,we get:$(b + c) 	imes (a + b + c) = b 	imes a + b 	imes c + c 	imes a - b 	imes c = b 	imes a + c 	imes a$.Now,take the dot product with $(a + b)$:$(a + b) \cdot (b 	imes a + c 	imes a) = a \cdot (b 	imes a) + a \cdot (c 	imes a) + b \cdot (b 	imes a) + b \cdot (c 	imes a)$.Using the properties of scalar triple products $[x y z] = x \cdot (y 	imes z)$:$a \cdot (b 	imes a) = [a b a] = 0$ (since two vectors are identical).$a \cdot (c 	imes a) = [a c a] = 0$.$b \cdot (b 	imes a) = [b b a] = 0$.$b \cdot (c 	imes a) = [b c a] = [a b c]$.Thus,the expression simplifies to $[a b c]$.

$(a + b) \cdot (b + c) \times (a + b + c) = $

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