Understanding Engineering Mathematics

x O i j k y z Figure 11.10Basic vectorsi,j,k. and this provides the link with the ‘arrows’ representation of vectors given in ea ...

Exercises on 11.8 Express the position vectors with the given endpoints in terms ofi,j,kvectors. (i) (3,−1, 2) (ii) (1, 0, 1) ...

x O y z P(x,y,z) OPr Figure 11.12Position vector of a pointP. Themagnitudeorlengthof a vectora=a 1 i+a 2 j+a 3 kis given by a=|a ...

Multiplication by a positive (negative)λleaves the direction ofaunchanged (reversed). −ais defined by(− 1 )a. Vectorsa=a 1 i+a 2 ...

(i) √ 21 2 (ii) 1 √ 21 (i+ 4 j− 2 k) (iii)− 3 i− 12 j+ 3 k (iv) 1 2 i+ 2 j+k (v) 5 2 i+ 10 j− 4 k 11.10 The scalar product of ...

q d F Figure 11.13Force at an angle to the direction of motion. wire). This component isFcosθ. So, if we move the bead a distanc ...

Note the special case: a·a=a^21 +a 22 +a 32 =a^2 ≡|a|^2 =square of magnitude ofa In the above form of the scalar product it is n ...

a·b=(− 1 )( 0 )+ 2 ( 2 )+ 1 ( 3 )= 7 =abcosθ= √ (− 1 )^2 + 22 + 12 √ 02 + 22 + 32 cosθ = √ 6 √ 13 cosθ So cosθ= 7 √ 6 √ 13 givin ...

The notationa∧bis sometimes used. This vector product is often represented in the form of the array shown below, in order to aid ...

For the basis vectorsi,j,kwe find (Reinforcement Exercise 24) i×i=j×j=k×k= 0 i×j=k, j×k=i, k×i=j Geometrically the vector produc ...

11.12 Vector functions Avector functionis a vector which is a function of some variable, sayt. We writef(t) for a general vector ...

O A P r n a Figure 11.18Vector equation of a plane. is the general equation for a plane. Ifn=αi+βj+γkthen in terms of coordinate ...

Letf(t)=cbe a constant vector. Thenf(t+h)=c, and so from the above definition df dt =lim h→ 0 ( f(t+h)−f(t) h ) =lim h→ 0 ( c−c ...

Second and higher derivatives may be obtained in the obvious way by repeated differ- entiation. Thus, for example: d^2 f(t) dt^2 ...

is a vector tangential to the curve, pointing in the direction of motion of the particle and with magnitude equal to the magnitu ...

Verify d dt (f×g)= df dt ×g+f× dg dt for the two vectors: f=ti− 2 t^2 j+etkg=costi− 2 tj 11.14 Reinforcement 1.The free vector ...

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 o ...

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+ ...

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 ...

11.16 Answers to reinforcement exercises 1. (i) (ii) a (^) − c (iii) a^ + 2 b (iv) a (v) (a (^) − (^) b) 23 (a^ +^ b) (^12) a^ + ...