Question

A point p(7,3,1)" is attached to a frame Fnoa and is subjected to the following transformations. Find the coordinates of the point relative to the reference frame at end of all transformations. 1. Rotation of 90° about the z-axis. 2. Followed by a rotation of 90° about the y-axis 3. Followed by a translation of [4,-3,7].

Answer 1

We have point P= [7 3 1]

In 3D, we take it as P= [7 3 1 1]

Then we apply given transformation

Let

T_{​​​​​​1} = matrix of rotation of 90° about the z-axis.

T_{​​​​​​2}​​​ = Matrix of rotation of 90° about the y-axis.

T_{​​​​​​3}​​​ =Matrix of translation by [4 -3 7]

Hence T is matrix of all given transformation,

i.e. T = [T_{​​​​​​1}][T_{​​​​​​2}][T_{​​​​​​3}]
T = matrix of rotation of go about the Z-axis T = Cosgoo singoo oo o o o -singo cosgoo o ol -1 0 0 0 0 0 1 T2 = of goo matrix of rotation | Cosgoo o singoo o about y axis o ool -singoo o cosgoo o o oo Ta = matoix of translation by [4 -3 7] 4 -3 7 1 Now, T = T1 T2 T3 . T= 100 OO 10 toooooo Iooo 1 O O O O O 1o TOOOooo 1 4 -3 7 ) matrix product, T = To lo o to Hence, at point relative to the reference frame end of all transformation is, pt. pt = [p] [I]

looyo tooo 1 4 -3 7 1 Hence, point relative to the of all transformation reference is, frome at end

So ans is [3 4 4]