ENEL101 Problem set 2 Mathematical operations

Description

1. Given a vector of samples 𝑋 = [−100, −99, … ,99,100], create a vector 𝑌 calculated based
on element-­‐‑by-­‐‑element operations such that 𝑦2 = 0.001 𝑥2
‘ + 1. Construct matrix 𝑄 such
that
𝑄 =
𝑋
𝑌
𝑋 + 𝑌
2. The electric field intensity, 𝐸(𝑥), due to a ring of radius 𝑅 at any point 𝑥 along the
axis of the ring is given by:
𝐸 𝑥 = 9604.5
𝑅𝑥
𝑥2 + 𝑅2
3
2
Given R=6 cm and a row vector 𝑋 from elements ranging from 2 cm to 6 cm and spacing of
0.01 cm, use element-­‐‑by-­‐‑element operations to calculate a row vector 𝑌 whose elements are
𝑦2 = 𝐸(𝑥2). Then construct a matrix of
𝑄 = 𝑋
𝑌
Hint: use ‘format longg’ to see the significant digits (otherwise you’ll just see 0.0000)
3. The voltage 𝑉(𝑡) in Volt and the current 𝑖(𝑡) in Amp 𝑡 seconds after closing the switch in
an RC circuit are given by
𝑉 𝑡 = 24 1 − 𝑒E F
GH
𝐼 𝑡 = 24
𝑅
𝑒E F
GH
given 𝑅 = 3800 Ω and 𝐶 = 4000×10EN F. Create a row vector 𝑇, with values of times from 0
to 20 sec with spacing of 2 sec, and use it to create row vectors 𝑋 and 𝑌 such that 𝑥2 = 𝑉(𝑡2)
and 𝑦2 = 𝐼(𝑡2). Then construct a matrix of
𝑄 =
𝑇
𝑋
𝑌
4. Two vectors are given as
𝐮 = −1𝐢 + 2.5𝐣 + 0.5𝐤
𝐯 = +1𝐢 + 0.2𝐣 + 2𝐤
Compute the dot product as 𝑄 = 𝐮 ⋅ 𝐯.
5. Use cross() to compute the vector cross product as 𝑄 = 𝐮×𝐯, where 𝑄 is a row vector
and 𝐮 and v are two row vectors defined in question 4.
6. Use Matlab to show that the reciprocals of square numbers produce a convergent series
1
𝑛’
W
XY%
= 𝜋’
6
Do this by computing the following sums
𝑋 = 1
𝑛’
%[
XY%
, 𝑌 = 1
𝑛’
\[
XY%
, 𝑍 = 1
𝑛’
%[[
XY%
and placing your answers in matrix 𝑄 = 𝑋 𝑌 𝑍
7. Given two matrices
𝐴 =
1 −1 1
−1 1 −1
1 −1 1
and 𝐵 =
10 2 0
3 8 3
1 3 12
Determine 𝑄 = 𝐴𝐵𝐵_ where the superscript denotes transpose
8. Given two matrices
𝐴 =
1 −3 5
2 2 4
−2 0 6
and 𝐵 =
0 −2 1
5 1 −6
2 7 −1
Determine 𝑄 = 𝐴E%(𝐴 + 𝐵𝐴_)
9. Given matrix 𝐴 =
1 −3 5
2 2 4
−2 5 6
Let 𝐵 be the matrix formed by having each element of A inverted such that 𝐵 2,a = 1/ 𝐴 2,a
Then calculate 𝑄 = 𝐴𝐵.
10. The following linear set of equations are determined by applying mesh current analysis
to a circuit. Solve the linear set of equations using the \ operator in Matlab
−44𝑖% + 10𝑖’ + 16𝑖c = −20
10𝑖% − 43𝑖’ + 6𝑖c + 12𝑖d = 0
16𝑖% + 6𝑖’ − 30𝑖c + 8𝑖d = 12
12𝑖’ + 8𝑖c − 34𝑖d = −40
Let 𝑄 = [𝑖% 𝑖’ 𝑖c 𝑖d] be a row vector of the results.