Thread: Inner Product Spaces

1st question, how do you solve this.

"Find the angle between the diagonal of a cube and one of its edges."

2nd question, how to prove this.

"Prove that if u is orthogonal to v and w, then u is orthogonal to cv + dw for any scalars c and d."

thank you.

Originally Posted by test2k6

1st question, how do you solve this.

"Find the angle between the diagonal of a cube and one of its edges."

Let the vertices of the cube be (0,0,0) (1,0,0) (0,1,0) (0,0,1) (1,1,0) (0,1,1) (1,0,1) and (1,1,1).

Then the diagonal from (0,0,0) to (1,1,1) is in the direction (1,1,1), and
the edge from (0,0,0) to (1,0,0) is in the direction (1,0,0).

Now the cos(theta) = (1,1,1).(1,0,0)/[|(1,1,1)| |(1,0,0)| = 1/sqrt(3)
where theta is the angle getween the diagonal and the edge.

So: theta = arccos(1/sqrt(3)) ~= 0.955 radian or 54.7 degrees

RonL

Originally Posted by test2k6

2nd question, how to prove this.

"Prove that if u is orthogonal to v and w, then u is orthogonal to cv + dw for any scalars c and d."

thank you.

u is orthogonal to v and w means that u.v=0 and u.w=0.

Now u.(c v+d w) = c u.v + d u.w = 0,

so u is orthogonal to cv + dw

RonL