Problem 1

Find the distance between and .

Problem 2

Determine if and are orthogonal.

Problem 3

Suppose is orthogonal to and . Show that is orthogonal to every in .

Problem 4

Let , and . Determine whether the angle in the triangle is acute.

Problem 5

Let and . Show that is an orthogonal basis for , and express as a linear combination of the 's.

Problem 6

Let , and . Show that is an orthogonal basis for , and express as a linear combination of the 's.

Problem 7

Let and . Write as the sum of a vector in and a vector orthogonal to .

Problem 8

Let and . Compute the distance from to the line through and the origin.

Problem 9

Determine if the set of vectors is orthonormal. If not, normalize the vectors to produce an orthonormal set.

Problem 10

Let be an orthogonal matrix. Show that the rows of form an orthonormal basis of .

Problem 11

Let be the subspace spanned by the 's, and write as the sum of a vector in and a vector orthogonal to .

Problem 12

Find the closest point to in the plane spanned by and . Find the distance from to .

Problem 13

Let , , and .

(a). Let . Compute and

(b). Compute and

Problem 14

Let , , and .

(a). Find the Closest point in to

(b). Find the Distance from to

Problem 15

Let be the reflection on about the line . Find its matrix .