Question 1 :
Find the direction cosines of a vector whose direction ratios are
(i) 1 , 2 , 3 (ii) 3 , - 1 , 3 (iii) 0 , 0 , 7
Solution :
(i) x = 1, y = 2 and z = 3
|r vector| = r = √(x2 + y2 + z2) = √(12 + 22 + 32)
= √(1+4+9)
= √14
Hence direction cosines are ( 1/√14, 2/√14, 3/√14)
(ii) x = 3, y = -1 and z = 3
|r vector| = r = √(x2 + y2 + z2) = √(32 + (-1)2 + 32)
= √(9+1+9)
= √19
Hence direction cosines are ( 3/√19, -1/√19, 3/√19)
(iii) x = 0, y = 0 and z = 7
|r vector| = r = √(x2 + y2 + z2) = √(02 + 02 + 72)
= √49
= 7
Hence direction cosines are ( 0, 0, 1)
Question 2 :
Find the direction cosines and direction ratios of the following vectors.
(i) 3i vector - 4j vector + 8k vector
Solution :
x = 3, y = -4 and z = 8
|r vector| = r = √(x2 + y2 + z2) = √(32 + (-4)2 + 82)
= √(9+16+64)
= √89
Direction cosines :
(x/r, y/r, z/r)
x/r = 3/√89
y/r = -4/√89
z/r = 8/√89
Hence direction cosines are ( 3/√89, -4/√89, 8/√89)
Direction ratios :
Direction ratios are (3, -4, 8).
(ii) 3i vector + j vector + k vector
Solution :
x = 3, y = 1 and z = 1
|r vector| = r = √(x2 + y2 + z2) = √32 + 12 + 12)
= √(9+1+1)
= √11
Direction cosines :
(x/r, y/r, z/r)
x/r = 3/√11
y/r = 1/√11
z/r = 1/√11
Hence direction cosines are ( 3/√11, 1/√11, 1/√11)
Direction ratios :
Direction ratios are (3, 1, 1).
(iii) j vector
Solution :
x = 0, y = 1 and z = 0
|r vector| = r = √(x2 + y2 + z2) = √02 + 12 + 02)
= √1
= 1
Direction cosines :
(x/r, y/r, z/r)
x/r = 0/1 = 0
y/r = 1/1 = 1
z/r = 0/1 = 0
Hence direction cosines are ( 0, 1, 0)
Direction ratios :
Direction ratios are (0, 1, 0).
(iv) 5i vector - 3j vector - 48k vector
Solution :
x = 5, y = -3 and z = -48
|r vector| = r = √(x2 + y2 + z2) = √52 + (-3)2 + (-48)2
= √(25 + 9 + 2304)
= √2338
Direction cosines :
(x/r, y/r, z/r)
x/r = 5/√2338
y/r = -3/√2338
z/r = -48/√2338
Hence direction cosines are
(5/√2338, -3/√2338, -48/√2338)
Direction ratios :
Direction ratios are (5, -3, -48).
(v) 3i vector + 4j vector - 3k vector
Solution :
x = 3, y = 4 and z = -3
|r vector| = r = √(x2 + y2 + z2) = √32 + 42 + (-3)2
= √(9 + 16 + 9)
= √34
Direction cosines :
(x/r, y/r, z/r)
x/r = 3/√34
y/r = 4/√34
z/r = -3/√34
Hence direction cosines are
(3/√34, 4/√34, -3/√34)
Direction ratios :
Direction ratios are (3, 4, -3).
(vi) i vector - k vector
Solution :
x = 1, y = 0 and z = -1
|r vector| = r = √(x2 + y2 + z2) = √12 + 02 + (-1)2
= √(1 + 0 + 1)
= √2
Direction cosines :
(x/r, y/r, z/r)
x/r = 1/√2
y/r = 0/√2 = 0
z/r = -1/√2
Hence direction cosines are
(1/√2, 0, -1/√2)
Direction ratios :
Direction ratios are (1, 0, -1).
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