Three vectors A→, B→ and C→ add upto zero. Find which is false.

Since A→+B→+C→=0, the three vectors will have a plane that passes through them.( Since the three vectors form a triangle)A→+B→+C→=0 ⇒(A→+B→+C→)xB→=0xB→ (B→xB→=0) ⇒(A→xB→=B→xC→) .....(i) ⇒(A→xB→)xC→=(B→xC→)xC→ As B→xC→ is perpendicular to both B→ and C→ So (B→xC→)xC→ will be zero only if B→ and C→ are parallel. Considering (i) A→xB→=B→xC→ ⇒ (A→xB→).C→=(B→xC→).C→ As B→xC→ is always perpendicular to C→⇒ (B→xC→).C→=0 considering (i) A→xB→=B→xC→⇒(A→xB→)xC→=(B→xC)xC→ B→xC→ will be in a plane perpendicular to both B→ and C→ so (B→xC→)xC→ will be perpendicular to both B→xC→ and C→ so (B→xC→)xC→ must be in the plane of the traingle formed by A→, B→, C→. So(A→xB→)xC→ will be in that plane. If C2=A2+B2 ⇒ A→and B→ are perpendicular ⇒A→xB→=A→B→sin90°=A→B→ (A→xB→) and C→ are parallel ⇒(A→xB→).C→=A→B→C→cos0°=A→B→C→

Three vectors A→, B→ and C→ add upto zero. Find which is false.

Since A→+B→+C→=0, the three vectors will have a plane that passes through them.( Since the three vectors form a triangle)A→+B→+C→=0 ⇒(A→+B→+C→)xB→=0xB→ (B→xB→=0) ⇒(A→xB→=B→xC→) .....(i) ⇒(A→xB→)xC→=(B→xC→)xC→ As B→xC→ is perpendicular to both B→ and C→ So (B→xC→)xC→ will be zero only if B→ and C→ are parallel. Considering (i) A→xB→=B→xC→ ⇒ (A→xB→).C→=(B→xC→).C→ As B→xC→ is always perpendicular to C→⇒ (B→xC→).C→=0 considering (i) A→xB→=B→xC→⇒(A→xB→)xC→=(B→xC)xC→ B→xC→ will be in a plane perpendicular to both B→ and C→ so (B→xC→)xC→ will be perpendicular to both B→xC→ and C→ so (B→xC→)xC→ must be in the plane of the traingle formed by A→, B→, C→. So(A→xB→)xC→ will be in that plane. If C2=A2+B2 ⇒ A→and B→ are perpendicular ⇒A→xB→=A→B→sin90°=A→B→ (A→xB→) and C→ are parallel ⇒(A→xB→).C→=A→B→C→cos0°=A→B→C→

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

Three vectors $\vec{A}$ , $\vec{B}$ and $\vec{C}$ add upto zero. Find which is false.

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

$(\vec{A} \times \vec{B}) \times \vec{C}$ is non zero unless $\vec{B}$ and $\vec{C}$ are parallel.

$(\vec{A} \times \vec{B}) \cdot \vec{C}$ is non zero when $\vec{B}$ and $\vec{C}$ are parallel.

If $\vec{A}$ , $\vec{B}$ and $\vec{C}$ define a plane, $(\vec{A} \times \vec{B}) \times \vec{C}$ is in that plane.

$(\vec{A} \times \vec{B}) \cdot \vec{C} = | \vec{A} | | \vec{B} | | \vec{C} | \to C^{2} = A^{2} + B^{2}$ .

answer is B.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Since $\vec{A} + \vec{B} + \vec{C} = 0$ , the three vectors will have a plane that passes through them.( Since the three vectors form a triangle)

$\vec{A} + \vec{B} + \vec{C} = 0 \Rightarrow (\vec{A} + \vec{B} + \vec{C}) x \vec{B} = 0 x \vec{B} (\vec{B} x \vec{B} = 0) \Rightarrow (\vec{A} x \vec{B} = \vec{B} x \vec{C}) . . . . . (i) \Rightarrow (\vec{A} x \vec{B}) x \vec{C} = (\vec{B} x \vec{C}) x \vec{C} A s \vec{B} x \vec{C} i s p e r p e n d i c u l a r t o b o t h \vec{B} a n d \vec{C} S o (\vec{B} x \vec{C}) x \vec{C} w i l l b e z e r o o n l y i f \vec{B} a n d \vec{C} a r e p a r a l l e l . C o n s i d e r i n g (i) \vec{A} x \vec{B} = \vec{B} x \vec{C} \Rightarrow (\vec{A} x \vec{B}) . \vec{C} = (\vec{B} x \vec{C}) . \vec{C} A s \vec{B} x \vec{C} i s a l w a y s p e r p e n d i c u l a r t o \vec{C} \Rightarrow (\vec{B} x \vec{C}) . \vec{C} = 0 c o n s i d e r i n g (i) \vec{A} x \vec{B} = \vec{B} x \vec{C} \Rightarrow (\vec{A} x \vec{B}) x \vec{C} = (\vec{B} x C) x \vec{C} \vec{B} x \vec{C} w i l l b e i n a p l a n e p e r p e n d i c u l a r t o b o t h \vec{B} a n d \vec{C} s o (\vec{B} x \vec{C}) x \vec{C} w i l l b e p e r p e n d i c u l a r t o b o t h \vec{B} x \vec{C} a n d \vec{C} s o (\vec{B} x \vec{C}) x \vec{C} m u s t b e i n t h e p l a n e o f t h e t r a i n g l e f o r m e d b y \vec{A}, \vec{B}, \vec{C} . S o (\vec{A} x \vec{B}) x \vec{C} w i l l b e i n t h a t p l a n e . I f C^{2} = A^{2} + B^{2} \Rightarrow \vec{A} a n d \vec{B} a r e p e r p e n d i c u l a r \Rightarrow |\vec{A} x \vec{B}| = |\vec{A}| |\vec{B}| \sin 90 ° = |\vec{A}| |\vec{B}| (\vec{A} x \vec{B}) a n d \vec{C} a r e p a r a l l e l \Rightarrow (\vec{A} x \vec{B}) . \vec{C} = |\vec{A}| |\vec{B}| |\vec{C}| \cos 0 ° = |\vec{A}| |\vec{B}| |\vec{C}|$