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If either 
or
, then
. Is the converse true? Justify your answer with an example.
Take any parallel non-zero vectors so that.
It can now be observed that:
Hence, the converse of the given statement need not be true.
Find a unit vector perpendicular to each of the vector 
and
, where 
and
.
We have,
and
Hence, the unit vector perpendicular to each of the vectors and is given by the relation,
If a unit vector 
makes an angles
with 
with 
and an acute angle θ with
, then find θ and hence, the compounds of
.
Let unit vector have (a1, a2, a3) components.
⇒ 
Since is a unit vector, 
.
Also, it is given that makes angleswith with , and an acute angle θ with
Then, we have:
Hence,
and the components of are
.
Find λ and μ if 
.
On comparing the corresponding components, we have:
Hence, 
Given that 
and
. What can you conclude about the vectors
?
Then,
(i) Either 
or
, or 
(ii) Either or, or 
But, 
and 
cannot be perpendicular and parallel simultaneously.
Hence, or.
Let the vectors 
given as
. Then show that 
We have,
On adding (2) and (3), we get:
Now, from (1) and (4), we have:
Hence, the given result is proved.
Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and
C (1, 5, 5).
The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and
C (1, 5, 5).
The adjacent sides
and
of ΔABC are given as:
Area of ΔABC 
Hence, the area of ΔABC
Find the area of the parallelogram whose adjacent sides are determined by the vector 
.
The area of the parallelogram whose adjacent sides are 
is
.
Adjacent sides are given as:
Hence, the area of the given parallelogram is
.
Let the vectors 
and 
be such that 
and
, then
is a unit vector, if the angle between 
and is
(A) 
(B) 
(C) 
(D) 
It is given that
.
We know that
, where 
is a unit vector perpendicular to both and 
and θ is the angle between and.
Now, is a unit vector if
.
Hence, is a unit vector if the angle between and is.
The correct answer is B.
Area of a rectangle having vertices A, B, C, and D with position vectors 
and 
respectively is
(A) 
(B) 1
(C) 2
(D) 
The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:
The adjacent sides
and 
of the given rectangle are given as:.png)
⇒???AB−→−×BC−→???=2
Now, it is known that the area of a parallelogram whose adjacent sides are 
is
.
Hence, the area of the given rectangle is
The correct answer is C.
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, if 
and
. 
