Transcribed Text
1. Do any two of a - c.
Consider the lines given by the equations x − 2y = 2 and 2x + y = −2.
a. Find the angle between the lines.
b. Find vector-parametric equations for each of the lines.
c. Find the point where the lines intersect.
2. Do any two of a - c.
Let u =
1
1
1
1
, v =
1
0
1
1
, andw =
1
1
0
1
a. Find the angle θ between u and v.
b. Solve the equation 2u + 4w + 5x = −3v for x.
c. Find the components of w that are, respectively, parallel to and perpendicular to v.
3. Consider the following system of linear equations
2x + y + z = 1
x + y = 1
x + 2y − z = 1
a. Find all solutions, if any, of this system.
b. Determine whether the vectors
2
1
1
1
1
2
, and
1
0
−1
are linearly dependent.
4. As in 2, let u =
1
1
1
1
, v =
1
0
1
1
, andw =
1
1
0
1
. In addition, let x =
1
.
a. Determine whether x ∈ Span{u, v, w}.
b. Determine whether u, v and w are linearly dependent or independent.
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