Class 12 RD Sharma Solutions – Chapter 23 Algebra of Vectors – Exercise 23.9
Question 1: Can a vector have direction angles 45°, 60°, and 120°.
Solution:
We know that if l, m and n are the direction cosines and , and are the direction angles then,
=>
=>
=>
Also,
=> l2 + m2 + n2 = 1
=>
=>
=> As LHS = RHS, the vector can have these direction angles.
Question 2: Prove that 1,1 and 1 can not be the direction cosines of a straight line.
Solution:
Given that, l=1, m=1 and n=1.
We know that,
=> l2 + m2 + n2 = 1
=> 12 + 12 + 12 = 1
=> 3 ≠ 1
Thus, 1, 1 and 1 can never be the direction cosines of a straight line.
=> Hence proved.
Question 3: A vector makes an angle of with each of x-axis and y-axis. Find the angle made by it with the z-axis.
Solution:
We know that if l, m and n are the direction cosines and , and are the direction angles then,
=>
=>
Let be the angle we have to calculate.
We know that,
=> l2 + m2 + n2 = 1
=>
=> n2 = 1 – 1
=> n2 = 0
=>
=>
=>
=>
Question 4: A vector is inclined at equal acute angles to x-axis, y-axis and z-axis. If = 6 units, find .
Solution:
Given that
=>
=> l = m = n = p (say)
We know that,
=> l2 + m2 + n2 = 1
=> p2 + p2 + p2 = 1
=> 3p2 = 1
=>
The vector can be described as,
=>
=>
=>
Question 5: A vector is inclined to the x-axis at 45° and y-axis at 60°. If units, find .
Solution:
Given that and
We know that,
=> l2 + m2 + n2 = 1
=>
=>
=>
=>
=>
=>
The vector can be described as,
=>
=>
=>
Question 6: Find the direction cosines of the following vectors:
(i):
Solution:
The direction ratios are given as 2, 2 and -1.
Direction cosines are given as,
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=>
(ii):
Solution:
The direction ratios are given as 6, -2 and -3.
Direction cosines are given as,
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=>
=>
(iii):
Solution:
The direction ratios are given as 3, 0 and -4.
Direction cosines are given as,
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=>
=>
Question 7: Find the angles at which the following vectors are inclined to each of the coordinates axes.
(i):
Solution:
The given direction ratios are: 1,-1,1.
Thus,
=>
=>
=>
=>
=>
(ii):
Solution:
The given direction ratios are: 0,1,-1.
Thus,
=>
=>
=>
=>
=>
=>
(iii):
Solution:
The given direction ratios are: 4, 8, 1.
Thus,
=>
=>
=>
=>
=>
Question 8: Show that the vector is equally inclined with the axes OX, OY and OZ.
Solution:
Let
Thus,
=>
Thus the direction cosines are: , and
=>
Thus,
=>
=> Thus, the vector is equally inclined with the 3 axes.
Question 9: Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ,, .
Solution:
Let the vector be equally inclined at an angle of .
Then the direction cosines of the vector l, m, n are: , and
We know that,
=> l2 + m2 + n2 = 1
=>
=>
=>
=> Thus the direction cosines are: , , .
Question 10: If a unit vector makes an angle with , with and an acute angle with, then find \theta and hence the components of .
Solution:
The unit vector be,
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=>
Given that is unit vector,
=>
=>
=>
=>
=>
=>
=>
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Question 11: Find a vector of magnitude units which makes an angle of and with y and z axes respectively.
Solution:
Let l, m, n be the direction cosines of the vector .
We know that,
=> l2 + m2 + n2 = 1
=>
=>
=>
=>
Thus vector is,
=>
=>
=>
Question 12: A vector is inclined at equal angles to the 3 axes. If the magnitude of is , find .
Solution:
Let l, m, n be the direction cosines of the vector .
Given that the vector is inclined at equal angles to the 3 axes.
=>
We know that,
=> l2 + m2 + n2 = 1
=>
=>
Hence, the vector is given as,
=>
=>
=>
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