Q. 13
Find the value of λ for which the four points with position vectors 
and 
are coplanar.
Answer :
Given :
Let, A, B, C & D be four points with given position vectors
To Find : value of λ
Formulae :
1) Vectors :
If A & B are two points with position vectors 
,
Where,
then vector 
is given by,
2) Scalar Triple Product:
If
Then,
3) Determinant :
Answer :
For given position vectors,
Vectors 
are given by,
………eq(1)
………eq(2)
………eq(3)
Now, for vectors
= -2(2 + 2λ) – 0 + 2(4)
= - 4 - 4λ + 8
= 4 – 4λ
………eq(4)
Four points A, B, C & D are coplanar if and only if
………eq(5)
From eq(4) and eq(5)
4 – 4λ = 0
4λ = 4
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The angle between two vectors 
and 
with magnitudes 
and 4, respectively, and 
is
True and False
If 
, then the vectors 
and 
are orthogonal.
Fill in the blanks
The vectors 
are the adjacent sides of a parallelogram. The acute angel between its diagonals is ____________.
Fill in the blanks
If 
is any non-zero vector, then 
equals ______.
True and False
If 
and 
are adjacent sides of a rhombus, then 
.
True and False
The formula 
is valid for non-zero vectors 
and 
.
Fill in the blanks
The vector 
bisects the angle between the non-collinear vectors 
and 
if ________
If 
and 
are two vectors such that 
then prove that vector 
is perpendicular to vector 
Mark the correct alternative in each of the following:
If θ is an acute angle and the vector 
is perpendicular to the vector 
then θ =
RD Sharma - Volume 2
Mark the correct alternative in each of the following:
is equal to
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