Download MATH 2120 W13 Review 1 1 1. Find the three angles of the triangle

MATH 2120 W13 Review 1
1
1. Find the three angles of the triangle △ABC with vertices A(1, 0), B(3, 6), C(−1, 4).
Ans. 45◦ , 45◦ , 90◦ .
2. For what values of b are the vectors (−6, b, 2) and (b, b2 , b) orthogonal?
3. Find a unit vector that is orthogonal to both (1, 1, 0) and (1, 0, 1).
Ans.
√1 (1, −1, −1).
3
4. Find the acute angle between the lines 2x − y = 3 and 3x + y = 7. Ans. 45◦ .
5. Find the vector projection of v = (2, −1, 4) onto u = (0, 1, 12 ).
Ans.
1
(2, −1, 4).
21
6. Find the distance from the point (−2, 3) to the line 3x − 4y + 5 = 0. Ans.
7.
13
.
5
a) Find the angle between the planes x + y + z = 1 and x − 2y + 3z = 1.
b) Find the equation of the line of intersection of the planes.
   


x
1
5
Ans. a) 72◦ b)  y  =  0  + t  −2  .
z
0
−3
8. Find the
√ distance between the parallel planes 10x + 2y − 2z = 5 and 5x + y − z = 1.
Ans. 63 .
9. Find the equation of the plane through the point (6, 3, 2) and perpendicular to the
vector v = (−2, 1, 5).
Ans. −2x + y + 5z = 1.
10. Find the equation of the plane that passes through the point (−1, 2, 1) and contains
the line of intersection of the planes x + y − z = 2 and 2x − y + 3z = 1.
Ans. x − 2y + 4z = −1.
11. Find the point at which the line x = 3 − t, y = 2 + t, z = 5t intersects the plane
x − y + 2z = 9.
Ans. (2, 3, 5).
12. Prove that u · v = 14 ∥u + v∥2 − 14 ∥u − v∥2 for all vectors u, v in Rn .
13. Prove that u + v and u − v are orthogonal in Rn if and only if ∥u∥ = ∥v∥.
14. Solve the system of equations using either Gaussian or Gauss-Jordan elimination.
w + x + 2y + z
w−x−y+z
x+y
w+x+z
Ans. No solution.
=
=
=
=
1
0
−1
2
MATH 2120 W13 Review 1
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15. For what values of k, if any, will the sytems have a) no solution, b) a unique solution,
and c) infinitely many solutions?
x + y + kz = 1
x + ky + z = 1
kx + y + z = −2
16. Determine by inspection(without reducing the matrix) whether the linear system with
the augmented matrix below has a unique solution, infinitely many solutions, or no
solution.


1 2 3 4 5 6 7
 7 6 5 4 3 2 1 
8 8 8 8 8 8 8
Ans. No solution.

  
 
1
1
1
3





0 ,
1 , and
1 .
17. Show that R is spanned by the vectors
0
0
1
 
 
3
0



4  geometrically.
0
and
18. Describe the span of the vectors
1
0
Ans. Line in R3 through the points (0, 0, 0) and (3, 4, 1).
 
 
 
0
2
2
19. Determine if the vectors u =  1  , v =  1  , and w =  0  are linearly
2
3
1
independent or not. If the vectors are linearly dependent, find a dependence relation
among the vectors.
Ans. Lin. dependent, −u + v = w.
20. Prove that two vectors in Rn are linearly dependent if and only if one is a scalar
multiple of the other.
(
)
(
)
1 2
−1 0
21. Given A =
and B =
, solve the equation 2(A − B + 2X) =
3 4
1 1
3(X − B) for X.
(
)
(
)
1 2
a b
22. If A =
and B =
, find conditions on a, b, c, d such that AB = BA.
3 4
c d
Ans. 3b = 2c, a = d − c.
23. Show that for m × n matrices A and B, (A + B)T = AT + B T .
24. Show that for any matrix A, the matrix AAT is symmetric.