Q. 154.0( 4 Votes )
Find the value of λ for which the points A(3, 2, 1), B(4, λ, 5), C(4, 2, -2) and D(6, 5, -1) are coplanar.
Ans. λ = 5
Answer :
Given :
Points A, B, C & D are coplanar where,
A ≡ (3, 2, 1)
B ≡ (4, λ, 5)
C ≡ (4, 2, -2)
D ≡ (6, 5, -1)
To Find : value of λ
Formulae :
1) Position Vectors :
If A is a point with co-ordinates (a1, a2, a3)
then its position vector is given by,
2) Vectors :
If A & B are two points with position vectors 
,
Where,
then vector 
is given by,
3) Scalar Triple Product:
If
Then,
4) Determinant :
Answer :
For given points,
A ≡ (3, 2, 1)
B ≡ (4, λ, 5)
C ≡ (4, 2, -2)
D ≡ (6, 5, -1)
Position vectors of above points are,
Vectors 
are given by,
………eq(1)
………eq(2)
………eq(3)
Now, for vectors
= - 1(9) – (2 - λ).(7) – 4(3)
= - 9 – 14 + 7λ – 12
= 7λ – 35
………… eq(4)
But points A, B, C & D are coplanar if and only if
………… eq(5)
From eq(4) and eq(5)
7λ – 35 = 0
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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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