Midterm 2 Flashcards
Find the length of u = (-1,1,3,-5)
sqrt(1^2 + 1^2 + 3^2 + 5^2) = 6
What are the two other ways to say “Find the length of a vector u”?
1) Find the Euclidean norm/length of u
2) Find ||u||
Find a unit vector in the direction u = (1, -5, 3, 2)
= u/||u||
= (1,-5,3,2)/sqrt(1^2 + 5^2 + 3^2 + 2^2)
Find the Euclidean distance between u = (-1,0,3) and v = (2,1,2) or find d(u, v)
= ||v - u||
= sqrt((2 - (-1))^2 + (1 - 0)^2 + (2 - 3)^2)
= sqrt(11)
Given u = (2, 0, -1) and v = (4, 0, 1), which of the following are true?
a) u and v are parallel
b) u is a unit vector
c) ||v|| = 5
d) u * v = 7
a) False, they’re not multiples of each other
b) False, the length of u is not 1
c) False, ||v|| = sqrt(17)
d) True, u * v = 8 - 1 = 8
Find the angle between vectors u = (5, 3) and v = (4, -1).
Use u * v = ||u|| ||v|| cos(theta) and rearrange for theta
When are two vectors orthogonal and what does this mean?
Two vectors are orthogonal or perpendicular if the dot product is zero
Are any of the given vectors orthogonal to one another?
u = (2,-1,2,1)
v = (1,1,-1,1)
w = (1,1,3,1)
u * v = 0 (yes)
u * w = 8 (no)
v * w = 0 (yes)
Given u = (2,-1,3) and a = (1,0,2) find,
a) the vector component of u along a
b) the vector component of u orthogonal to a
a) (1,0,2)
b) (2/5, -1, -1/5)
What is the point-normal form of a equation of the
plane? explain how to get the 3 values.
a(x - x0) + b(y - y0) + c(z - z0) = 0
(a, b, c) = normal vector of the plane
(x, y, z) = arbitrary point on the plane
(x0, y0, z0) = a known point on the plane
Find the equation of the plane that’s parallel to the plane 4x - 2y + 5z = 6 that goes through the point (-1, 0, 8).
4(x - (-1)) - 2(y - 0) + 5(z - 8) = 0
What is the equation of the plane in standard form?
ax + by + cz = d
What is the parametric equations of a line L given (x, y, z) = (x0, y0, z0) + t(a, b, c)
x = x0 + ta
y = y0 + tb
z = y0 + tc
Find the parametric equations of the line L containing the point p = (1, -2, 3) and perpendicular to the plane 2x - y + 3z = 5
If L is perpendicular to the equation then it is parallel to the normal vector n = (2, -1, 3)
x = 1 + 2t
y = -2 - t
z = 3 + 3t
Which of the following statements are true?
a) The planes 3x + y - 2z = 7 and 6(x - 2) + 2y - 4(z + 1) = 0 are parallel
b) The planes 7x - y + 3z = 6 and 2x + 7y + 4z = 0 are perpendicular
c) The lines x = 3 + 2t, y = 5 - t, z = 6 + 3t and x = 7 - 2t, y = 9 + t, z = 1 - 3t are parallel
d) The line x = 3 + 2t, y = 5 - t, z = 6 + 3t is perpendicular to the plane 2x - y 3z = 5
a) True
b) False
c) True
d) False
Find the cross product of (6,3,7) x (2,9,0)
= (-63, 14, 48)
Find the area of the parallelogram determined by the vectors u = (2, -1, 3) and v = (1, 5, 1)
u x v = sqrt(378)
What makes up a vector space?
1) The objects {(u1, u2, u3, …, un)}
2) Closed under addition u + v
3) Closed under scalar multiplication ku
For which of the following sets will vector space axiom 6 hold? ku belongs to V.
a) Vectors in the form (a, b) where a and b are positive real numbers
b) Vectors of the form (a, 3, b)
c) Upper triangular matrices
d) 2 x 2 matrices with a determinant of 0
a) No
b) No
c) Yes
d) Yes
For which of the following sets will vector space axiom 5 hold? u + (-u) = (-u) + u = 0
a) Vectors in the form (a, b, 3)
b) Vectors of the form (a, b, 0)
c) Diagonal matrices
d) 2x2 matrices where all the entries are positive
e) symmetric matricies
a) No
b) Yes
c) Yes
d) No
e) Yes
Let V be the set of all points on the line y = x + 1. Does this form a vector space?
No,
v = (5, 6)
u = (1, 2)
u + v = (6, 8) not in V
For which of the following sets will vector space property 1 hold? u + v is in V
a) (a, b, 1)
b) (a, b, 0)
c) Diagonal matrices
d) 2x2 matrices, all positive
a) No
b) Yes
c) Yes
d) Yes
When is w a sub space
a) closed under addition u + v is in W
b) closed under scalar multiplication ku is in W
Which of the following are subspaces of the indicated vector space?
a) Is the set of all 2x2 matrices A of the form|-1 b| a subspace of M22?
| 0 c|
b) (a, b, 0) with b = 3a a subspace of R^3
c) {a0 + a1x + a2x^2 | a0 + a1 = 0} a subspace of P2
a) No
b) Yes
c) Yes