Understanding Engineering Mathematics

18.Find the scalar products of the following pairs of vectors and state if any of the three
pairs are perpendicular

(i) i−j,3i+ 4 j+ 5 k (ii) 4i+j− 3 k,−i+ 3 j− 7 k

(iii) 3i+j+ 4 k,2i− 2 j−k

19.Find the angles between the pairs of vectors

(i) a=−j+kb= 3 i+ 4 j+ 5 k

(ii) a=i−j− 2 kb= 2 i+j+ 3 k

(iii) a= 2 i+j−kb=i+ 2 j+k

20.Suppose the axesOxyzare such thatOxpoints East,Oypoints North,Ozpoints
vertically upwards. Evaluate the scalar products of the vectorsaandbin each of the
following cases:
(i)ais of magnitude 3 and points SE.bhas length 2 and points E.
(ii) ais a unit vector pointing NE.bis of magnitude 2 and points vertically upwards.
(iii)ais of unit magnitude and points NE.bis of magnitude 2 and points W.

21.Ifa=i−jb=−j+ 2 k, show that

(a+b)·(a− 2 b)=− 9

22.Show thati+j+kanda^2 i− 2 aj+kare perpendicular if and only ifa=1.

23.Determine the vector producta×bfor each of the pairs of vectors in Q18 and Q19.
In each case find unit vectors perpendicular to the two vectors.

24.Prove the results

i×j=kj×k=ik×i=j

i×i=j×j=k×k= 0

25.Find the vector products of the vectorsa,b

(i) a= 3 i+ 7 j+ 2 kb=i+ 3 j+k

(ii) a=i− 3 jb=− 2 i+ 5 j

(iii) a= 8 i+ 8 j−kb= 5 i+ 5 j+ 2 k

26.Ifa= 3 i− 2 j+k,b=i+j+kandc= 2 i+j− 3 kevaluate, using any short-cuts
you can:
(i) thetriple scalar products
a·b×ca·c×bb·a×b
(ii) thetriple vector products
a×(b×c)(a×b/×cb×(b×a)