If veca, vecb, vec c are three non-zero vectors and hatn is a unit vector perpendicular to vec c such that vec a =α vec b - hat n ,(α ≠ 0) and vec b ⋅ vec c =12, then | vec c ×( vec a × vec b )| is equal to :

Question

If veca, vecb, vec c are three non-zero vectors and hatn is a unit vector perpendicular to vec c such that vec a =α vec b - hat n ,(α ≠ 0) and vec b ⋅ vec c =12, then | vec c ×( vec a × vec b )| is equal to : (A) 9 (B) 15 (C) 6 (D) 12

Tardigrade Admin · Accepted Answer

hat n perp vec c vec a =α vec b - vec n vecb ⋅ vecc=12 vec a ⋅ vec c =α( vec b ⋅ vec c )- vec n ⋅ vec c veca ⋅ vecc=α( vecb ⋅ vecc) | vecc ×( veca × vecb)|=|( vecc ⋅ vecb) veca-( vecc ⋅ veca) vecb| =|( vec c ⋅ vec b ) vec a -α( vec b ⋅ vec c ) vec b | =|( vecc ⋅ vecb)|| vec a -α vec b | =12 ×(| vec n |) =12 × 1 =12

Q. If a,b,c are three non-zero vectors and n^ is a unit vector perpendicular to c such that a=αb−n^,(α=0) and b⋅c=12, then ∣c×(a×b)∣ is equal to :

9

15

6

12

n^⊥ca=αb−n b⋅c=12 a⋅c=α(b⋅c)−n⋅c a⋅c=α(b⋅c) ∣c×(a×b)∣=∣(c⋅b)a−(c⋅a)b∣ =∣(c⋅b)a−α(b⋅c)b∣ =∣(c⋅b)∣∣a−αb∣ =12×(∣n∣) =12×1 =12

Q. If $a, b, c$ are three non-zero vectors and $\overset{n}{^}$ is a unit vector perpendicular to $c$ such that $a = α b - \overset{n}{^}, (α \neq = 0)$ and $b \cdot c = 12$ , then $∣ c \times (a \times b) ∣$ is equal to :