Vectors and the Geometry of Space

Question

Determine whether the given vectors are orthogonal, parallel, or neither.

1. v=⟨1, 2,−1⟩ and w=⟨−10,−20,10⟩.

2. v=⟨5, 10,14⟩ and w=⟨10,−5,0⟩.

3. v=⟨10, 20,−30⟩ and w=⟨−30,20,10⟩.

Answer 1

Q1 For vectors to be parallel, they have to be linearly dependent. And to find that, you must divide v₁/w₁ , v₂/w₂ and v₃/w₃ and see if they are equal

1. (1/-10)=-1/10
2. (2*-20)=-1/10
3. (-1*10)=-/10

For vectors to be orthogonal, their products (V * W) must equal to 0

(1*-10)+(2*-20)+(-1*10)= -10-20-10=-40

its not equal to zero, so it is not orthogonol

Q2

1. 5/10= 1/2
2. 10/-5= -2
3. 14/0= DNE

Not parallel

(5*10)+(10*-5)+(14*0)=50-50=0

vectors are orthogonal

Q3

1. 10/-30=1/3
2. 20/20=1
3. -30/10=-3

Not parallel

(10*-30)+(20*20)+(-30*10)= -300+400-300=400

Not Orthogonal, so neither.