Homework 3

MAT 281E

1. (15 pts.) What are two possible unit vectors these are orthogonal to both u = (2, 0, 1)

and v = (0, 1, 1).

2

Name:

Number: Homework 3

MAT 281E

November 1, 2019

2. (20 pts.) u = (4, 2, 3, 1) and a = (2, −2, 1, −1)

a) projau

b) Find the vector compenent of u orthogonal to a.

3

Name:

Number: Homework 3

MAT 281E

November 1, 2019

3. (15 pts.) Find the distance between parallel planes.

a) 3x − y − z = 5 and 6x − 2y − 2z = 8.

b) −x + y + 2z = 0 and −3x + 3y + 6z = 0.

4

Name:

Number: Homework 3

MAT 281E

November 1, 2019

4. (10 pts.) x = (t + 1)(4, 6) + t(−1, 0)

Use this equation of a line to find a point on the line and a vector parallel to the line.

5

Name:

Number: Homework 3

MAT 281E

November 1, 2019

5. (10 pts.) v = (2, 0, −3)

Find vector and parametric equations of the plane in R3

that passes through the origin

and orthogonal to v.

6

Name:

Number: Homework 3

MAT 281E

November 1, 2019

6. (30 pts.) Consider the linear systems

6 4 −2

3 2 −1

−6 −4 2

x1

x2

x3

=

0

0

0

and

6 4 −2

3 2 −1

−6 −4 2

x1

x2

x3

=

4

2

−4

a) Find a general solution of the homogeneous system.

b) Confirm that x1 = 1, x2 = 0, x3 = 1 is a solution of the nonhomogeneous system.

c) Use the results in parts (a) and (b) to find a general solution of the nonhomogeneous

system.

d)Check your result in part (c) by solving the nonhomogeneous system directly.

7

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