- #1
olliebear
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Here are some questions that I received on my test. I got most of these wrong but I got a couple points here and there. Here are the questions I had trouble with. At the bottom on the questions I tried to solve them but I'm not sure if i did them correctly. Please try and help me because I have a final coming up this week and I want to learn how to solve these questions.
1. Consider the three vectors u=(4,-1,-5) , v=(1,-4,1) and w= (1,1,-2)
a) compute the scalar triple product of u,v,w.
my attempt:
u * (v X w)
4 -1 -5
1 -4 1
1 1 -2
=4(7) + 1(-3) -5(5)
= |0|
=0
b) What can you deduce about the vectors u,v,w, supposing they have the same initial point?
my attempt: That the answer would be different because it would be multiplying with different multiples…?
2. Considers the points A(1,0,1) B(1,2,3) C(-1,0,2)
a) find the angle between the vectors AB and AC
my attempt:
cos=AB*AC/||AB|| ||AC||
=2/(8)^1/2 (5)^1/2
= 0.3162
=cos-1(0.3162)
=71.56
b) Find a vector equation of the line through A and B
my attempt:
A=(1,0,1)
B= (1,2,3
?
3. Considers the L1(line 1) with symmetric equation (x-1)/-1 = (y+2)/2 =z/3
and the L2(line 2) parallel to v2=(1,0,-1) and throughout the point P2(0,1,0)
a) Find the direction vector v1 for the line L1 and give a point P1 on L1
my attempt:
p1 (1,1, 1/3)
v1 (1,1,1)
b) Find a parametric equation of the line L2
my attempt:
v2= (1,0,-1)
P2= (0,1,0)
L2 should equal {t, 1, -1}
c) show that the lines L1 and L2 are skew lines
my attempt:
|x1x2 * (V1 X V2)| \ ||V1 X V2||
=(0,1,0) * ( -1, -2, -1) / 6^1/2
= -2/6^1/2 d) Find a unit vector u orthogonal to both v1 and v2
my attempt:
v1 (1,1,1)
v2 (1,0, -1)
v3 ( , , )
v1 X v2
1 1 1
1 0 -1
u=( -1,0,-1)
e) find the orthogonal projection P1P2 on u
(p1p2 * u/ ||u|| ) u
=(-2/2^1/2, 0, -2/2^1/2)
f) deduce the distance d=||Proju P1P || between the lines L1 and L24. If u and v are vector in n-space, Simplify: ( u+v) * (u-v)
my attempt:
=uu –uv + uv - vv
=||u||^2 - ||v||^2
b) use your previous result to show that the parallelogram defined by u and v is a rhombus if and only if its diagonals are perpendicular.
A rhombus has 4 sides that are equal and this would prove it.
1. Consider the three vectors u=(4,-1,-5) , v=(1,-4,1) and w= (1,1,-2)
a) compute the scalar triple product of u,v,w.
my attempt:
u * (v X w)
4 -1 -5
1 -4 1
1 1 -2
=4(7) + 1(-3) -5(5)
= |0|
=0
b) What can you deduce about the vectors u,v,w, supposing they have the same initial point?
my attempt: That the answer would be different because it would be multiplying with different multiples…?
2. Considers the points A(1,0,1) B(1,2,3) C(-1,0,2)
a) find the angle between the vectors AB and AC
my attempt:
cos=AB*AC/||AB|| ||AC||
=2/(8)^1/2 (5)^1/2
= 0.3162
=cos-1(0.3162)
=71.56
b) Find a vector equation of the line through A and B
my attempt:
A=(1,0,1)
B= (1,2,3
?
3. Considers the L1(line 1) with symmetric equation (x-1)/-1 = (y+2)/2 =z/3
and the L2(line 2) parallel to v2=(1,0,-1) and throughout the point P2(0,1,0)
a) Find the direction vector v1 for the line L1 and give a point P1 on L1
my attempt:
p1 (1,1, 1/3)
v1 (1,1,1)
b) Find a parametric equation of the line L2
my attempt:
v2= (1,0,-1)
P2= (0,1,0)
L2 should equal {t, 1, -1}
c) show that the lines L1 and L2 are skew lines
my attempt:
|x1x2 * (V1 X V2)| \ ||V1 X V2||
=(0,1,0) * ( -1, -2, -1) / 6^1/2
= -2/6^1/2 d) Find a unit vector u orthogonal to both v1 and v2
my attempt:
v1 (1,1,1)
v2 (1,0, -1)
v3 ( , , )
v1 X v2
1 1 1
1 0 -1
u=( -1,0,-1)
e) find the orthogonal projection P1P2 on u
(p1p2 * u/ ||u|| ) u
=(-2/2^1/2, 0, -2/2^1/2)
f) deduce the distance d=||Proju P1P || between the lines L1 and L24. If u and v are vector in n-space, Simplify: ( u+v) * (u-v)
my attempt:
=uu –uv + uv - vv
=||u||^2 - ||v||^2
b) use your previous result to show that the parallelogram defined by u and v is a rhombus if and only if its diagonals are perpendicular.
A rhombus has 4 sides that are equal and this would prove it.
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