Q. 203.7( 3 Votes )

Show that the four points A, B, C and D with position vectors and respectively are coplanar.

OR

The scalar product of the vector with a unit vector along the sum of vectors and is equal to one. Find the value of λ and hence find the unit vector along

Answer :

The position vectors of points


, , and


The points A, B, C and D are coplanar if the vectors , and are coplanar


Vectors , and are coplanar of


Let us first write the vectors , and











Now let us find the value of


represents the determinant



Expand the determinant along the first row






Hence vectors , and are coplanar and hence points A, B, C and D are coplanar


OR


, and


Given that the dot product of with unit vector along is 1




To find a unit vector along we have to divide by


Hence unit vector along





Take dot product of this unit vector along with






Square both sides,


(2 + λ)2 + 40 = (6 + λ)2


40 = (6 + λ)2 – (2 + λ)2


40 = (6 + λ + 2 + λ)(6 + λ – 2 – λ)


40 = (8 + 2λ)4


10 = 2(4 + λ)


5 = 4 + λ


λ = 1


Hence λ = 1.


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