Understanding Engineering Mathematics

7.A straight rod is held with one end in the corner of a room. If it makes angles of 60°

and 45°with the lines of intersection of the floor and the walls, find the angle that it

makes with the vertical.

8.Find the direction cosines of the vectorsOPfor each of the pointsP in Q5.

9.Determine the acute angles between the position vectors defined by the pairs of points

in Q6.

10.Express in terms ofi,j,kthe position vectors with endpoints

(i) (2,−1, 2) (ii) (1, 2, 3) (iii) (−1,−2,−3) (iv) (2,−3, 4) (v) (2u, 3v, 4w)

11.Calculate the magnitude and direction for each of the position vectors:

(i) a= 3 i (ii) b=i+j (iii) c= 3 i− 3 j

(iv) d=

√

3 i+j (v) e=i+ 2 j (vi) f=i+j+k

(vii) g=−i−j+ 2 k (viii) h= 2 i+ 2 j+ 2 k

Find unit vectors in the direction of each vector.

12.For the vectors in Q11 find

(i) a+b (ii) c−g (iii) a+b+h

(iv) 2a− 3 b− 3 f (v) 4a+ 2 b− 3 c+d+e−f+g− 3 h

13.For the vectors in Q11 determine the vectorxsuch that:

(i) 2x− 3 a+b= 0 (ii) a+b− 2 c+ 3 x=d

(iii) 2x− 2 f= 3 x+ 2 g (iv) 3x+ 2 e−f=g+h−x

14.The following describe position vectors with respect to the basis vectorsi,j,k. Write
down the component forms of the vectors.
(i)ris in thexyplane, has length 3 units and bisects the angle betweeniandj.
(ii) ris in theyzplane, has length 2 units and makes angle 30°withjand 60°withk.
(iii)ris the diagonal of the unit cube in the first octant with sidesi,j,k.

15.Vectorsa,b,chave magnitudes 1, 2, 3 respectively and the angle betweenaandb
is 60°, betweenbandc 45 °and betweenaandcis 105°. Find all possible scalar
products betweena,b,c.

16.Prove the results

i·i=j·j=k·k= 1

i·j=i·k=j·k= 0

17. aandbare perpendicular to each other with magnitudes 2 and 4 respectively. If
    x= 2 a−bandy= 3 a− 4 b,findx·y.Ifz=λa−bdetermineλsuch thatyandz
    are perpendicular.