“Determine wether u and v are orthogonal, parallel or neither, where: u = <1,2,-3> and v = <-4/3,-8/3,4>.

I am thinking they’re parallel.”

The Dot Product

u • v = (1)(-4/3) + (2)(-8/3) + (-3)(4), and

u • v = √(1+4+9)√(16/9 + 64/9 + 16) cos(θ),

where θ is the angle between the vectors.

cos(θ) = ((1)(-4/3) + (2)(-8/3) + (-3)(4))

/(√(1+4+9)√(16/9 + 64/9 + 16))

= (-4/3 -16/3 - 36/3)/√((14)(80/9 + 144/9))

= (-56/3)/√((14)(80/9 + 144/9))

= -56/√((14)(80 + 144))

= -56/56

= -1

θ = 180° so the vectors are parallel.

==

“1)Find the vector z = -2u + (1/2)v - 3w,

given u = <-1,3,2>, v = <6,-2,-2> and w = <5,0,-5>.

So i am thinking that i have to first find the internal and terminal point to find the component”

u = <-1,3,2>

v = <6,-2,-2>

w = <5,0,-5>

z = -2u + 1/2v - 3w

-2u = <2,-6,-4>

(1/2)v = <3,-1,-1>

-3w = <-15,0,15>

z = <(2+3-15,-6-1+0,-4-1+15>

z = <-10,-7,10>