HOW TO FIND THE DIRECTION COSINES OF A VECTOR WITH GIVEN RATIOS

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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