Vectors and planes

Question

1) Find u.v

2) Are u and v orthogonal?

3) Is u a unit vector?

4) Is v a unit vector?

5) Is u parallel to the plane?

6) Is v parallel to the plane?

Answer 1

1) u = (2, -2, 1) and v = (1, 2, 2)

u = < 2, -2, 1 > and v = < 1, 2, 2 >

u . v = (2)(1) + (-2)(2) + (1)(2)

= 2  - 4 + 2

u . v = 0

2) Since u . v = 0, u,v are orthogonal.

3) u = < 2, -2, 1 >

|u| = √(4 + 4 + 1) = √9 = 3

Unit vector = u/|u| = < 2, -2, 1 >/3

Yes, u is unit vector.

4) v = < 1, 2, 2 >

|v| = √(1 + 4 + 4) = √9 = 3

Unit vector = v/|v| = < 1, 2, 2 >/3

Yes ,v is unit vector.

5) Plane 2x + y -2z = -2

n = < 2, 1, -2 >

Vector u = < 2, -2, 1>

u.n = < 2, -2, 1>.< 2, 1, -2 >

     =4-2-2

    = 0

Plane is parallel to u.

6) Plane 2x + y -2z = -2

< 2, 1, -2 >

Vector v = < 1, 2, 2>

v.n = < 1,2, -2>.< 2, 1, -2 >

     =2+2-4

      = 0

Plane is  parallel to v.

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