vectors u v and w

Question

given u=(2,0,-3) v=(3,1,2) and w=(0,-2,1)

(a) which pairs of the following vectors are orthogonal?

(b) find the angle between a pair of these vectors which are not orthogonal

(c) express (1,5,3) as linear combination of u,v,and w

Answer 1

Two vectors u and v whose dot product is zero are orthogonal. 

 

and the dot product is x_1*x₂ + y_1*y₂ + z₁*z₂

 

Let's check u and v from your problem.

 

u*v = 2*3 + 0*1 - 3*2 = 0. So this vectors are orthogonal.

 

Let's check u and w

 

u*w = 2*0 + 0*(-2) + (-3)*1 = -3

Not orthigonal

 

Now we need to find an angle between those vectors.

u*w = |u|*|w|*cos(∠uw) 

 

so as we know |u| = √(x²+y²+z²) = √(4+0+9) = √13

 

Could you find |w|  and find the cosine of the angle from this information? Or do you need more help?

 

I also didn't check if v and w are orthogonal. Could you do that?

 

As for (c) part, you could take x of u, y of v and z of w. And you get

2x+3y+0z=1

0x+1y-2z =5

and what would be the 3rd equation here?

 

Could you solve the system now?