For non-zero vectors a,b,c, |(a×b).c|=| a || b || c | holds, if and only if

Question

For non-zero vectors a,b,c, |(a×b).c|=| a || b || c | holds, if and only if (A) a.b=0,b.c=0 (B) b.c=0,c.a=0 (C) c.a=0,a.b=0 (D)  a ⋅ b = b ⋅ c = c ⋅ a =0

Tardigrade Admin · Accepted Answer

Given, |( a × b ) ⋅ c |=| a || b || c | ⇒ | a | b | sin θ hat n ⋅ c |=| a | b || c | ⇒ | a || b || c || sin θ ⋅ cos α|=| a || b || c | ⇒ | sin θ| ⋅| cos α|=1 ⇒ θ=(π/2) and α=0 ∴ a perp b and c | hat n i.e. a perp b and c perpendicular to both a and b

Q. For non-zero vectors $a, b, c, ∣ (a \times b) . c ∣ = ∣ a ∣∣ b ∣∣ c ∣$ holds, if and only if

$a . b = 0, b . c = 0$

$b . c = 0, c . a = 0$

$c . a = 0, a . b = 0$

$a \cdot b = b \cdot c = c \cdot a = 0$

Q. For non-zero vectors a,b,c,∣(a×b).c∣=∣a∣∣b∣∣c∣ holds, if and only if

a.b=0,b.c=0

b.c=0,c.a=0

c.a=0,a.b=0

a⋅b=b⋅c=c⋅a=0

Given, ∣(a×b)⋅c∣=∣a∣∣b∣∣c∣ ⇒∥a∥b∣sinθn^⋅c∣=∣a∥b∣∣c∣ ⇒∣a∣∣b∣∣c∣∣sinθ⋅cosα∣=∣a∣∣b∣∣c∣ ⇒∣sinθ∣⋅∣cosα∣=1⇒θ=2π​ and α=0 ∴a⊥b and c∥n^ i.e. a⊥b and c perpendicular to both a and b

Q. For non-zero vectors $a, b, c, ∣ (a \times b) . c ∣ = ∣ a ∣∣ b ∣∣ c ∣$ holds, if and only if

$a . b = 0, b . c = 0$

$b . c = 0, c . a = 0$

$c . a = 0, a . b = 0$

$a \cdot b = b \cdot c = c \cdot a = 0$