QA-9_3

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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

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