Homework 3 MAT 281E

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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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