Page No 454:
Find, if and.
Find a unit vector perpendicular to each of the vector and, where and.
Hence, the unit vector perpendicular to each of the vectors and is given by the relation,
If a unit vector makes an angleswith 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:
Given that and. What can you conclude about the vectors?
(i) Either or, or
(ii) Either or, or
But, and cannot be perpendicular and parallel simultaneously.
Let the vectors given as . Then show that
On adding (2) and (3), we get:
Now, from (1) and (4), we have:
Hence, the given result is proved.
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 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 sidesand of ΔABC are given as:
Area of ΔABC
Hence, the area of ΔABC
Page No 455:
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:
⇒AB→×BC→=2Now, 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.