Higher Engineering Mathematics, Sixth Edition

Scalar and vector products 283

Now try the following exercise

Exercise 113 Further problemson vector

products

In problems 1 to 4, determine the quantities

stated when

p= 3 i+ 2 k,q=i− 2 j+ 3 kand

r=− 4 i+ 3 j−k.

1. (a)p×q (b)q×p
   [(a) 4i− 7 j− 6 k(b)− 4 i+ 7 j+ 6 k]


2. (a)|p×r| (b)|r×q|
   [(a) 11.92 (b) 13.96]


3. (a) 2p× 3 r(b)(p+r)×q
   [
   (a)− 36 i− 30 j− 54 k
   (b) 11 i+ 4 j−k

]

4. (a)p×(r×q)(b)( 3 p× 2 r)×q
   [
   (a)− 22 i−j+ 33 k
   (b) 18 i+ 162 j+ 102 k

]

5. For vectorsp= 4 i−j+ 2 kand
   q=− 2 i+ 3 j− 2 kdetermine: (i)p•q
   (ii) p×q (iii) |p×q| (iv) q×p and
   (v) the angle between the vectors.
   ⎡
   ⎢
   ⎣

(i)− 15 (ii)− 4 i+ 4 j+ 10 k

(iii) 11. 49 (iv) 4 i− 4 j− 10 k

(v) 142. 55 ◦

⎤

⎥

⎦

6. For vectors a=− 7 i+ 4 j+^12 kand b= 6 i−
   5 j−k find (i) a•b (ii) a×b (iii) |a×b|
   (iv) b×a and (v) the angle between the
   vectors. ⎡

⎢

⎣

(i)− 6212 (ii)− 112 i− 4 j+ 11 k

(iii) 11. 80 (iv) 112 i+ 4 j− 11 k

(v) 169. 31 ◦

⎤

⎥

⎦

7. Forces of(i+ 3 j),(− 2 i−j),(i− 2 j)newtons
   act at three points having position vectors of
   ( 2 i+ 5 j),4jand(−i+j)metres respectively.
   Calculate the magnitude of the moment.
   [10Nm]


8. A force of( 2 i−j+k)newtons acts on a line
   through pointPhaving co-ordinates (0, 3, 1)
   metres. Determine the moment vector and its

magnitude about pointQhaving co-ordinates

(4, 0,−1) metres.

[

M=( 5 i+ 8 j− 2 k)Nm,

|M|= 9 .64Nm

]

9. A sphere is rotating with angular velocityω
   about thez-axis of a system, the axis coincid-
   ing with the axis of the sphere. Determine the
   velocity vector and its magnitude at position
   (− 5 i+ 2 j− 7 k)m, when the angular velocity
   is(i+ 2 j)rad/s. [
   υ=− 14 i+ 7 j+ 12 k,
   |υ|= 19 .72m/s

]

10. Calculate the velocity vector and its magni-
    tude for a particle rotating about thez-axis
    at an angular velocity of( 3 i−j+ 2 k)rad/s
    when the position vector of the particle is at
    (i− 5 j+ 4 k)m.
    [6i− 10 j− 14 k,18.22m/s]

26.4 Vector equation of a line

The equationof a straightlinemay bedetermined, given

that it passes through the pointAwith position vector

arelative toO, and is parallel to vectorb.Letrbe the

position vector of a pointPon the line, as shown in

Fig. 26.10.

O

a

A

P

b

r

Figure 26.10

By vector addition,OP=OA+AP,

i.e.r=a+AP.

However, as the straight line throughAis parallel to the

free vectorb(free vectormeans one that has the same