Let a^{\to},b^{\to},c^{\to} be three vectors in the xyz- space such that a^{\to} \times b^{\to} = b^{\to} \times c^{\to} = c^{\to} \times a^{\to} \neq0.If A, B, C are points with position vectors a ^{\to},b^{\to} ,c^{\to} respectively, then the number of possible positions of the centroid of triangle ABC is

KVPY SB SX Exam 2011 Question Paper

Question : 18 of 120

Marks: +1, -0

Let

a↖{→},b↖{→},c↖{→}

be three vectors in the xyz- space such that

a↖{→} × b↖{→} = b↖{→} × c↖{→} = c↖{→} × a↖{→} ≠0.

If A, B, C are points with position vectors

a ↖{→},b↖{→} ,c↖{→}

respectively, then the number of possible positions of the centroid of triangle ABC is

Solution:

a↖{→} × b↖{→} + b↖{→} × c↖{→} =0

similarly

b↖{→} +c↖{→}=λ_2a↖{→}

a↖{→} +c↖{→}=λ_1b↖{→}

b↖{→} +a↖{→}=λ_3c↖{→}

Hence

a↖{→}+b↖{→} +c↖{→}=0

⇒ only 1 position of centroid

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