1. Consider the vectors x =< 4,6,2 >, y =< 8,-8,0 > and z =<7, 3,2 > in R3 (a) M
ID: 2900263 • Letter: 1
Question
1. Consider the vectors x =< 4,6,2 >, y =< 8,-8,0 > and z =<7, 3,2 > in R3
(a) Manually determine (x) x (y).
(b) Use the with(VectorCalculus): package and the DotProduct(. . . ); and CrossProduct(. . . ); com-
mands to determine z . ( (x) x (y) ) State both the commands used and the answer in your response.
(c) Referring to your answer in (b), what is the relationship between the vector determined by (x) x (y)
and the vector z?
What does this imply about the relationship between x, y and z?
for (b) and (c) use maple.
Explanation / Answer
a) (4,6,2)x(8,-8,0) = (6*0-(-8)*2,-(4*0-8*2),-8*4-6* 8) = (16,16,-80)
b)
x:=<4,6,2>
y:=<8,-8,0>
z:=<7,3,2>
DotProduct(z,CrossProduct(x,y))
Return 0
Manual check : (7,3,2).(16,16,-80) = 7*16+3*16-2*80 = 0
c) It means that (x) x (y) is perpendicular to z since the dot product is zero.
Since the cross product gives a perpendicular vector to the plan spanned by x and y, and z is perpendicular to this vector , it means that z is in the plan !
In other word x,y,z are linearly dependent.
Which can be verified by checking the determinant if you want ( not asked )
4 8 7
6 -8 3
2 0 2
det = 4*(-8*2) -6*8*2+2*(8*3+8*7) = 0
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