Question

ā,b,c and d are the position vectors of four coplanar points such that (ā-d).(b-c)=(b-d).(c-ā)=0. Show that the point d represents the orthocentre of the triangle with ā, b and c as its vertices.​

Answer 1

Answer:

Since (a−d).(b−c)=0, ∴DA.CB=0

∴AD⊥BC

Since (b−d).(c−a)=0,∴DB.AC=0

∴BD⊥CA.

Then D is the intersection of the altitudes through A and B.

Therefore, D is the orthocentre of the triangle ABC.

Step-by-step explanation:

hope it helps uhh..

Answer 2

Since (a−d).(b−c)=0, ∴DA.CB=0

∴AD⊥BC

Since (b−d).(c−a)=0,∴DB.AC=0

∴BD⊥CA.

Then D is the intersection of the altitudes through A and B.

Therefore, D is the orthocentre of the triangle ABC.