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is a unit vector in the XY-plane, then
Here,
θ
is the angle made by the unit vector with the positive direction of the
x
-axis.
Therefore, for
θ
= 30°:
Solution :
Given points are
Hence, the scalar components and the magnitude of the vector joining the given points are respectively
Hence, the girl’s displacement from her initial point of departure is
then is it true that
Justify your answer.
Solution :
.png)
) is a unit vector.
Solution :
x (
Hence, the required value of
x
is ± 1/√3.
Question 6. Find a vector of magnitude 5 units and parallel to the resultant of the vectors
Solution :
Given: Vectors
.png)
be the resultant of
.png)
and
.png)
Hence, the vector of magnitude 5 units and parallel to the resultant of vectors
Question 7. If
find a unit vector parallel to the vector
Solution :
Given: Vectors
Question 8. Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear and find the ratio in which B divides AC.
Solution :
The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).
Hence, point B divides AC in the ratio 2 : 3.
.
It is given that point R divides a line segment joining two points P and Q externally in the ratio 1: 2. Then, on using the section formula, we get:
Therefore, the position vector of point R is
.
Position vector of the mid-point of RQ =
Hence, P is the mid-point of the line segment RQ.
Question 10. Two adjacent sides of a parallelogram are
Find the unit vector parallel to its diagonal. Also, find its area.
Solution :
Adjacent sides of a parallelogram are given as:
Hence, the area of the parallelogram is 11√5 square units.
Question 11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
Solution :
Let a vector be equally inclined to axes OX, OY, and OZ at angle
α
.
Then, the direction cosines of the vector are cos
α
, cos
α
, and cos
α
.
Hence, the direction cosines of the vector which are equally inclined to the axes are
Find a vector
.png)
which is perpendicular to both
Question 13. The scalar product of the vector
.png)
with a unit vector along the sum of vectors
is equal to one. Find the value of λ.
Solution :
Hence, the value of
λ
is 1.
Question 14. If are mutually perpendicular vectors of equal magnitudes, show that the vector
Hence, the vector (
if and only if
.
Solution :
Question 16. Choose the correct answer:
If
θ
is the angle between two vectors
Solution :
Let
θ
be the angle between two vectors
Therefore, option (B) is correct.
Question 17. Choose the correct answer:
Let
Hence,
is:
(A) 0
(B) -1
(C) 1
(D) 3
Solution :
Therefore, option (C) is correct.
Question 19. If θ be the angle between any two vectors
when
θ is equal to:
(A) 0
(B) π/4
(C) π/2
(D) π
Solution :
Let
θ
be the angle between two vectors
Therefore, option (B) is correct.
