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

Опубликовано: 19 Июль 2026
на канале: Math Master - Impetus Gurukul

118

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
Ans: d
Sol.
Given that |(a ⃗×b ⃗ )∙c ⃗ |=|a ⃗ ||b ⃗ ||c ⃗ |
⇒|(|a ⃗ |)| b ⃗|sin⁡θ n ̂∙c ⃗ |=|a ⃗ ||b ⃗ ||c ⃗ |
⇒|a ⃗ ||b ⃗ ||c ⃗ ||sin⁡θ ||cos⁡α |=|a ⃗ ||b ⃗ ||c ⃗ |
⇒|sin⁡θ ||cos⁡α |=1⇒θ=π/2,α=0
⇒a ⃗ is perpendicular to b ⃗ and c ⃗ is parallel to n ̂
⇒a ⃗ is perpendicular to b ⃗ and c ⃗ is perpendicular to both a ⃗ and b ⃗
⇒a ⃗∙b ⃗=0,b ⃗∙c ⃗=0,c ⃗∙a ⃗=0
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The scalar A ⃗∙{(B ⃗+C ⃗ )×(A ⃗+B ⃗+C ⃗ )}equals
(a) 0 (b) [A ⃗B ⃗C ⃗ ]+[B ⃗C ⃗C ⃗ ]
(c) [A ⃗B ⃗C ⃗ ] (d) None of these
Ans: a
Sol.
A ⃗∙{(B ⃗+C ⃗ )×(A ⃗+B ⃗+C ⃗ )}
=A ⃗∙{B ⃗×A ⃗+B ⃗×B ⃗+B ⃗×C ⃗+C ⃗×A ⃗+C ⃗×B ⃗+C ⃗×C ⃗ }
=A ⃗∙{-A ⃗×B ⃗+0 ⃗+B ⃗×C ⃗-A ⃗×C ⃗-B ⃗×C ⃗+0 ⃗ } (∵a scalar triple product in which a vector is repeated has the value zero)

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