Problem 2 Given the vectors u2i6jk or vik or w8j7k or Com

Problem 2: Given the vectors

u=2i-6j+k [or <2,-6,1>]

v=-i+k [or <-1,0,1>]

w=8j-7k [or <0,8,-7>]

Compute each of the following quantities using the rules of vector algebra. I’m not particular about which notion you use – I like the pointy brackets (less writing,) but your text has a preference for the i,j,k notion.

a) -6 (w-u+2v)

b) [(1/(||3v-2w||)] u

c) (u+v) . (-2w) [Note that the middle is a “dot product dot” – there’s no such thing as “multiplying” vectors!]

d) Find the angle between the vectors u and v (degrees)

e) Find a vector in the direction of vector v, but whose magnitude is the same as ||w||

Solution

u=2i-6j+k

v=-i+k

w=8j-7k

a) -6(w-u +2v) = -6((0,8,-7) - ( 2, -6,1) +2(-1,0,1) )

=-6( -4 , 14 , -7)

= (24, -84 , 42)

= 24i -j84 +42k

b) [(1/(||3v-2w||)] u

(3v -2w) = 3( 2, -6 , 1) -2( 0, 8 -7) = (6, -34 , 17)

||3v -2w|| = sqrt(6^2 +34^2 +17^2) = 38.48

[(1/(||3v-2w||)] u = (2i -6j +k)/38.48 = 0.05i - j0.155 +0.026k

c) (u+v)(-2w)

(u+v) = i - 6j +2k

Dot product(u+v) (-2w) = ( 1, -6 , 2)(0, -16 , 14) = 0 +96+28 = 124

d) anlge between u and v:

dot product u.v = -2+1 = -1

||u|| = sqrt(2 +36+1) = sqrt39

||v|| = sqrt(1+1) = sqrt2

cosx = u.v/||u||.||v|| = -1/srt(39*2) = -1/8.83

x = 83.50 deg