Understanding Engineering Mathematics

5.(i)

√

2 (ii)

√

5 (iii)

√

5(iv)

√

14 (v) 5 (vi) 2

6.(i)

√

14 (ii) 2

√

3 (iii)

√

35 (iv)

√

22 (v)a

√

6

7. 60 °


8. (i)

(

1

√

2

,

1

√

2

, 0

)

(ii)

(

2

√

5

, 0 ,−

1

√

5

)

(iii)

(

−

2

√

5

, 0 ,

1

√

5

)

(iv)

(

−

2

√

14

,

3

√

14

,

1

√

14

)

(v)

(

0 ,

4

5

,

3

5

)

(vi)

(

1

√

2

,

1

2

,

1

2

)

9.(i) 74. 5 ° (ii) 75. 04 ° (iii) 61. 87 ° (iv) 54. 74 ° (v) 60°

10.(i) 2i−j+ 2 k (ii)i+ 2 j+ 3 k (iii)−i− 2 j− 3 k (iv) 2i− 3 j+ 4 k
(v) 2ui+ 3 vj+ 4 wk

11. (i) |a|=3,ˆa=i (ii) |b|=

√

2,bˆ=

1

√

2

(i+j)

(iii) |c|= 3

√

2,ˆc=

1

√

2

(i−j) (iv) |d|=2,dˆ=

√

3

2

i+

1

2

j

(v) |e|=3,eˆ=

1

√

5

i+

2

√

5

j (vi) |f|=

√

3,ˆf=

1

√

3

i+

1

√

3

j+

1

√

3

k

(vii) |g|=

√

6,gˆ=−

1

√

6

i−

1

√

6

j+

2

√

6

k

(viii) |h|= 2

√

3,hˆ=

1

√

3

i+

1

√

3

j+

1

√

3

k

The coefficients of the unit vectors give the direction cosines that define the directions

of the vectors.

12. (i) 4i+j (ii) 4i− 2 j− 2 k (iii) 6i+ 3 j+ 2 k

(iv) − 6 j− 3 k (v) (

√

3 − 2 )i+ 6 j− 5 k

13.(i)

1

2

( 8 i−j) (ii)

2 +

√

3

3

i− 2 j (iii) − 6 k (iv) −

1

2

j+

5

4

k

14.(i)

3

√

2

i+

3

√

2

j (ii)

√

3 j+k (iii) i+j+k

15.Betweenaandbthe scalar product is 1
Betweenbandcthe scalar product is 3

√

2

Betweenaandcthe scalar products is− 0 .78 to two decimal places

17. x·y=88,λ=−

16

3

18.(i) − 1 (ii) 20 (iii) 0, so these are orthogonal

19.(i) 84. 26 ° (ii) 123. 06 ° (iii) 60°

20.(i) 3

√

2 (ii) 0 (iii) −

√

2