Understanding Engineering Mathematics

27.For the pairs of vector functionsf(t),g(t)verify the product rules:

d

dt

(f.g)=f·

dg

dt

+

df

dt

·g

d

dt

(f×g)=

df

dt

×g+f×

dg

dt

(i)f=t^2 i+etj+sintkg=ti+t^2 j+ 2 k

(ii) f=cos 2ti+sin 2tj+ 3 kg=ti+ 2 j− 3 t^2 k

11.15 Applications

1.Vector methods can be used in geometrical applications. The following provide some
examples for you to try.

(i) The pointP divides the line segmentABin the ratiom:n.Ifa,b

are respectively the position vectors ofAandBwith respect to an

originO, determine the position vector of the pointPrelative toO.

(ii) Use the scalar product to prove Pythagoras’ theorem

(iii) Use the scalar product to prove the cosine rule

(iv) Adjacent sides of a triangle represent vectorsaandb. Show that the

area of the triangle is^12 |a×b|.

2.There are countless examples of use of vectors in mechanics of course. The following
give some typical examples.

(i) Assuming the triangle rule for addition of vectors show that any set

of forces represented by arrows forming a closed polygon with all

sides like directed is in equilibrium (i.e. sums to zero).

(ii) Themomentortorqueof a forceFacting at the pointP about the

originOis defined to beM=r×Fwhereris the position vector

ofP. Describe the magnitude and direction ofM.

Determine the torque of the forceF=i+ 2 j+ 3 kapplied at the point

(−1,−1, 2) about the points

(a) (0, 0, 0) (b) (3, 2,−1) (c) (1, 1, 1)

(iii) The position vector of a particle at timetis given by

r=(t+ 1 )i−t^2 j+etk

Determine the velocity and acceleration at timet.