30.11.2020
ES 202 HOMEWORK 1
(You may use any software such as MATHEMATICA, MATHCAD, MAPLE,
MATLAB, etc. to check your results )
DEADLINE FOR SUBMISSION: Dec. 14, 2020
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1. a) Write the following in indicial form: i) Force F , ii) position or displacement vector
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r , iii) velocity v , iv) acceleration a , v) dot product of two vectors, vi) cross product of
two vectors, vii) kronecker delta, viii) permutation symbol  , ix) stress, x) strain;
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b) If vectors u  ( 1,1,2) and v  (2,2,1) are given, calculate i)  ij u i v j =? ii)
1
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w i   ijk u j v k  ? , write w  ? , iii) E= p i q i where p i  a ij u j , q i  a ij v j , and
2
.
a ij  u i v j
2. Determine whether the given vectors of the indicated space are linearly dependent or
independent. Do they form a complete basis for the corresponding vector space? Give
your reasons.
a) (-1,0,1), (1,1,0),(-1,-1,1) in R3.
b) (3, -1,3), (-2,2,1), (2,1,-1) in R3 .
1
1
c) 4, (2x- ), ((x+1)2 - ) for P2 (x ) in [-1,1].
2
4
3. a) Check whether the planes x - y + z = 2 and -x - y = 3 are orthogonal.
b) For what values of ‘c’ the planes x +y -z = 2 and c x - y - z = 6 are
orthogonal?
c) Find the angle between the straight lines x – 3y = 2 and -2x + y = 3.
4. Consider the plane S with the equation x1+3x2+x3=1 and the line L with the parametric
equation: x1 = 1+2t ; x2 = -1+t ; x3 = 2+t. Let P be the intersection point of L and S.
a) Find the coordinates of the point P.
b) Find the parametric equations of the line L1
which passes through the point P and perpendicular
to S.
c) Find the equation of the intersection line of two
planes S and x1+2 x2- x3=0.
x1+3x2+x3=1
x1+2 x2- x3=0
5. Find the l1, l2 and l norms of the following vectors of R6 .
a) u = (1, -3,-5, -9.2, 1, 2)
b) v = (0, -7.6, 3.3, 2.5, 1, -5.1).
6. a) Find the projected vector for b = (1, -2, 5) on the subspace spanned by the vectors
(1, 2, - 1) and (1, - 1, 2).
b) Find the projected vector for the vector (0, 0, 1) on the plane 2x – 3y - z = 0 in R3.
c) Find the projected vector for the vector (0, 1, -1, 0) on the subspace (hyperplane)
x1 –x2 + x3 +x4 = 0 in R4 .
7. a) Using Gram – Schmidt procedure, from the following polynomial base functions
(BF’s) : ( 1, x , x2) obtain the following orthogonal base functions, i ’s
( i = 1 to 3) in the interval 0  x  1
1
1
1  1 ;  2    x ; 3   x  x 2 .
2
6
b) Compute the inner product of each i by itself that is
di = < i , i > =  i
2
1
1
0
0
   i  i dx    i2 dx , i = 1 to 3
and show that
1
1
; d3 =
.
12
180
c) Approximate the following triangular function “f”
f = 3(1 – 2 x) for 0  x  0.5 and f = 0 for 0.5  x  1
in 0  x  1.0 using the base functions you have found in part “a” and by
retaining three terms, that is,
f  c1 1 + c2 2 + c3 3
where
d1 = 1 ; d 2 =
ci 
 f , i 
; i =1 to 3
 i,i 
and compare the approximate function values with the exact in 0  x  1.0 by
drawing a figure.
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8. Considering the plane curve (in xy plane) C: r  ( t sin t , t cos t ) ; 0  t  2 . a) Sketch
C. b) Compute its length  from 0 to 2.