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course Mth 277
6/11 1
1) 9.3.4 Find v dot u when v =<1,-5,0> u =<0,-4,2>.The dot product of v dot u=0+20+0=20
2) 9.3.6 Find v dot w when v = 4i + j and w =3i + 2k.
v dot w= 12+0+0=12
3) 9.3.10 Determine whether v = 5i - 5j + 5k and w = 8i - 10j -2k are orthogonal.
Vectors are orthogonal if the dot product=0. 40+50-10=80 Therefore, these vectors are not orthogonal
4) 9.3.12 Let v = 4i - 2j + k and w = -2i + j - k. Evaluate (v dot w) * w.
First, we will do the expression inside the parenthesis.
v dot w=-8-2-1=-11 Then, we multiply this number by each component of the vector w.
-11*(8i-10j-2k)=88i-110j-22k
5) 9.3.16 Find the angle between v = 2i +3 k and w = -j + 4k.
v=<2,0,3> w=<0,-1,4>
magn v=sqrt(13)
magn w=sqrt(17)
v dot w=12
cos(theta)=12/(sqrt(13)*sqrt(17))
angle theta=36 degrees
6) 9.3.20 Find the scalar and vector projections of v = i - 2j onto w = j - 2k.
v=1,-2,0
w=0,1,-2
scalar= (v dot w)/(magn(w))=-2/sqrt(5)
vector proj=((v dot w)w/(magn(w))^2)=<0,1,-2>(-2/5)=<0,-2/5,4/5>
7) 9.3.24 Find two distinct unit vectors orthogonal to both v = i + 2j -2k and w = i + j - 2k.
First, we will do the cross product of the vector and then we divide by the magnitude to find the unit vector.
v x w=
1 2 -2
1 1 -2
-2 0 -1=<-2,0,-1>/(sqrt(5))=<-2/sqrt(5),0,-1/sqrt(5)>
I believe we can negate this unit vector to get another possible solution <2/sqrt(5),0,1/sqrt(5)>
8) 9.3.30 Find x so that v = 2i - xj + 3k and w = -2i + j + xk are orthogonal.
The dot product of these vectors need to equal zero.
So, -4-1+3x=0 x=5/3
9) 9.3.32 Give the direction cosines and direction angles of v = i - 4j
To find these, we do the arccos of each individual component divided by the magnitude of the vector
alpha=arccos(1/sqrt(17))=76 degrees
beta=arccos(4/sqrt(17))=14 degrees
gamma=arccos(0/sqrt(17))=0 degrees
10) 9.3.36(b,c,d) Let v = i - j + 4k and w = -i + 3j + 2k. Find cos(theta). Find s such that v is orthogonal to sv - w. Also find t such that v - tw is orthogonal to w
cos(theta) = v dot w/(magn(v)*magn(w))
cos(theta)=4/(sqrt(18)*sqrt(14))
(sv - w) dot v = 0
sv dot v=w dot v
s=(w dot v)/(v dot v)=4/18=2/9
(v - tw) dot w = 0
v dot w=tw dot w
t=(v dot w)/(w dot w)
t=4/14=2/7
11) 9.3.40 Find the work performed when a force F = (6/11)i - (2/11)j + (6/11)k is applied to an object moving along the line from P(3,5,-4) to Q(-4,-9,-11).
W = F dot PQ
PQ=-7,-14,-7
F dot PQ=(-42/11)+(-28/11)+(-42/11)=-112/11
&#This looks very good. Let me know if you have any questions. &#