Vector Algebra Question 74
Question: If $ \vec{a},\vec{b} $ and $ \vec{c} $ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?
Options:
A) $ \vec{a}+\vec{b}+\vec{c}=\vec{0} $
B) $ \vec{a}+\vec{b}+\vec{c}=unitvector $
C) $ \vec{a}+\vec{b}=\vec{c} $
D) $ \vec{a}=\vec{b}+\vec{c} $
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] Position vectors of vertices A, B and C are $ \overset{\to }{\mathop{a}},,\overset{\to }{\mathop{b}}, $ and $ \overset{\to }{\mathop{c}}, $ . $ \because $ triangle is equilateral.
$ \therefore $ Centroid and orthocenter will coincide. Centroid $ \equiv $ orthocenter position vector $ =\frac{1}{3}(\overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},+\overset{\to }{\mathop{c}},) $ $ \because $ given in question orthocenter is at origin. Hence $ \frac{1}{3}(\overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},+\overset{\to }{\mathop{c}},)=0 $ $ \overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},+\overset{\to }{\mathop{c}},=0 $
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