Let vec a = hat i + hat j + hat k and vec b = hat j - hat k . If vec c is a vector such that vec a × vec c = vec b and vec a . vec c =3, then vec a .( vec b × vec c ) is equal to :

Question

Let vec a = hat i + hat j + hat k and vec b = hat j - hat k . If vec c is a vector such that vec a × vec c = vec b and vec a . vec c =3, then vec a .( vec b × vec c ) is equal to : (A) -2 (B) -6 (C) 6 (D) 2

Tardigrade Admin · Accepted Answer

| vec a |=√3 ; vec a . vec c =3 ; vec a × vec b =-2 hat i + hat j + hat k , vec a × vec c = vec b Cross with veca. vec a ×( vec a × vec c )= vec a × vec b ⇒ ( vec a ⋅ vec c ) vec a - a 2 vec c = vec a × vec b ⇒ 3 vec a -3 vec c =-2 hat i + hat j + hat k ⇒ 3 hat i +3 hat j +3 hat k -3 vec c =-2 hat i + hat j + hat k ⇒ vec c =(5 hat i /3)+(2 hat j /3)+(2 hat k /3) ∴ vec a .( vec b × vec c )=( vec a × vec b ) . vec c =(-10/3)+(2/3)+(2/3)=-2

Q. Let a=i^+j^​+k^ and b=j^​−k^. If c is a vector such that a×c=b and a.c=3, then a.(b×c) is equal to :

-2

-6

6

2

∣a∣=3​;a.c=3; a×b=−2i^+j^​+k^,a×c=b Cross with a. a×(a×c)=a×b ⇒(a⋅c)a−a2c=a×b ⇒3a−3c=−2i^+j^​+k^ ⇒3i^+3j^​+3k^−3c=−2i^+j^​+k^ ⇒c=35i^​+32j^​​+32k^​ ∴a.(b×c)=(a×b).c =3−10​+32​+32​=−2

Q. Let $a = \hat{i} + \hat{j} + \hat{k}$ and $b = \hat{j} - \hat{k}$ . If $c$ is a vector such that $a \times c = b$ and $a . c = 3$ , then $a . (b \times c)$ is equal to :