TRANSFORMATIONS AND PROJECTIONS 45

can be determined aspp 12 ′× ′. The angle β between the planes PP P 123 ′′ ′ and Q 1 Q 2 Q 3 is now given

by cos β = p′ 3 · q 3. To orient the plane PP P 123 ′′ ′with respect to Q 1 Q 2 Q 3 at any desired angle θ, we

can rotate point P 3 ′ about PP 12 ′′ (or p 1 ′) through an angleθ−β as discussed in section 2.5.1.

Example 2.8.Given two triangular objects, S 1 {P 1 (0, 0, 1), P 2 (1, 0, 0), P 3 (0, 0, 0)} and S 2 {Q 1 (0,
0, 2), Q 2 (0, 2, 0), Q 3 (2, 0, 0)}, it is required that after assembly, point P 1 coincides with Q 1 and edge
P 1 P 2 lie on Q 1 Q 3. Determine the transformations if (i) S 1 is required to be in the same plane as S 2 and
(ii)S 1 is perpendicular to S 2.

Translation of P 1 P 2 P 3 to a new position PPP 1 * 2 * 3 * with P 1 to coincide with Q 1 is obtained by

P

P

P

P

P

P

T TT

1

*

2

*

3

*

1

2

3

= =

100 0

010 0

0 0 1 (2 – 1)

000 1

0011

1001

0001

=

0021

1011

0011

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

T ⎥

⎥⎥

⎥

T

It can be verified that P 2 lies on line Q 1 Q 3 and thus one does not need to perform step (b) above. It
is now required to determine the angle between the lamina PPP 1 2 3 and Q 1 Q 2 Q 3 which can be
obtained using step (c).

p

PP

PP

1

* 2

*

1

*

2

*

1

= * 22

• |– |

=

[(1 – 0), (0), (1 – 2), (1 – 1)]

1 + (– 1)

=

1

2

0 –^1

2

0

⎛

⎝

⎜

⎞

⎠

⎟

Similarly, p

PP

PP

p

PP PP

(^2) PP PP
*^3

• 1
  3 1
  3
  ^3


• 
  

  1 2 1
  3 1 2 1 *

  
  

• |– |

= (0 0 –1 0), =

(– ) (– )

| ( – ) ( – ) |

= (0 –1 0 0)

×

×

q

QQ

(^2) QQ
31
31 22

• 
  

  |– |

  
  

  [(2 – 0), (0), (0 – 2), (1 – 1)]
  2 + (– 2)

  
  

=^1

2

0 –

1

2

0

⎛

⎝

⎜

⎞

⎠

⎟

qq

QQ QQ

(^13) QQ QQ
31 21
31 21
= 0^1
2

• 
  

1

2

0 , =

( – ) ( – )

| ( – ) ( – ) |

=^1

3

1

3

1

3

0

⎛

⎝

⎜

⎞

⎠

⎟

×

×

⎛

⎝

⎜

⎞

⎠

⎟

Therefore,cos = =^1
3

1

3

1

3

0 (0 –1 0 0) = –^1

3

ββqp 33 ⋅ * = 125.26

⎛

⎝

⎜

⎞

⎠

⎟⋅⇒°

Angleβ (or, 180° – β) is the angle between the planes S 1 * and S 2 , and Q 1 Q 3 is the line about which

P 3 * is to be rotated to bring S 1 * to be either: (i) in plane with S 2 , or (ii) perpendicular to S 2.

The direction cosines of Q 1 Q 3 are given by

q 2 =

1

2

0 –

1

2

0 = ( 0)

⎛

⎝

⎜

⎞

⎠

⎟ nnnxyz

Following section 2.5.1, where rotation of a point about an arbitrary line is discussed, we shift Q 1 to
the origin, rotate line Q 1 Q 3 about the x-axis and then about y-axis

dn

n

d

n

x d

= 1 – =^1 z y

2

(^2) , = – 1, = 0