Vector math problems?

Question

The following are questions from my textbook which I cannot figure out:

Question 5:
Find a unit vector that is orthogonal to both u = (1,1,0) and v = (-1,0,1).

I tried turning the dot products into a linear system but it doesn't work out. I get a vector orthogonal to u, but it's not orthogonal to v.

Question 7:
Do the points A(1, 1, 1), B(-2, 0, 3) and C(-3, -1, 1) forrm the vertices of a right triangle?

According to the textbook the answer is "Yes" but if I use dot product then none of A, B or C seem to be orthogonal. Some help would be appreciated.

Plaustrarius · Accepted Answer

5)

In R^3 (3 dimensional space) you can find a vector orthogonal to two vectors by computing the cross-product between those two vectors.

u x v = vector orthogonal to both u and v

<1,1,0> x <-1,0,1> = <1,-1,1>

you can prove that this vector is orthogonal by dotting it with both u and v

now scale it down to make it a unit vector by dividing the vector by its length

(1/sqrt(3))<1,-1,1>

7)

first we'll need to find the vectors which make up the sides of this triangle

AB = <-2,0,3> - <1,1,1> = <-3,-1,2>

AC = <-3,-1,1> - <1,1,1> = <-4,-2,0>

BC = <-3,-1,1> - <-2,0,3> = <-1,-1,-2>

now you can either dot these vectors to prove orthogonality (AB dot BC will yield zero)

or you can show that the |AB|^2 + |BC|^2 = |AC|^2 (this order because AB and BC have smaller components than AC)

|AB| = sqrt(9+1+4) = sqrt(14)

|BC| = sqrt(1+1+4) = sqrt(6)

|AC| = sqrt(16+4+0) = sqrt(20)

14 + 6 = 20

so this is a right triangle

hope that helped!

Vector math problems?

1 Answer