Math 120 Quiz 3
Time Allowed: 1 hour
Total Marks: 20 7 March 2024
Quiz 3
1. (5 points)
Assume there is a parallelogram-shaped finite
sheet of metal with area A, with one corner at
the origin, and its two lengths spanned by the
vectors (1, 2, 0) and (−2, 4, 4) (unit of distance is
meters). This sheet is charged with a charge den-
sity of σ = +1 C/m2 .
Assuming there is a particle with charge q = −1 C
far away from the sheet of metal at (7.5, −1, 10),
the attractive force the charge experiences is given
by F = K|q||Q|
d2
, where d is the distance of the par-
ticle from the sheet of paper, Q = σA is the total
charge on the sheet of metal, and K is some constant.
Find F in terms of K.
2. (3 points)
The set of all real-valued continuous and twice-differentiable functions, denoted by V , is a real vec-
tor space with the usual addition and scalar multiplication given by (f + g)(x) = f (x) + g(x) and
(kf )(x) = k(f (x)). Let S ⊆ V be the set of all functions f such that −2f ′′ + 3f ′ + 5f = 0. Show that
S is a subspace of V .
3. (3 points)
Let Pn denote the set of polynomials with real coefficients and of degree exactly equal to n, where n ≥ 0
is an integer. Under the normal addition and scalar multiplication for polynomials, for what values of
n is Pn a real vector space? Justify your answer.
4 (2+3 points)
√
(i) Let u and v be two vectors in R3 , with |u| = 3, |v| = 6 and |v+u| = 63. Find u · v.
(ii) For any two vectors u and v in R3 , show that v · (u × v) = 0.
5 (4 points)
−3 1
Express as a product of elementary matrices.
2 2
Bonus (3 points)
You have studied that given two vectors v = (v1 , v2 , v3 ) and w = (w1 , w2 , w3 ) in R3 , the projection of v
onto w is given by projw v = ( v·w w
|w| ) |w| . Consider the line L(t) = tw through the origin (−∞ < t < ∞).
Show that the point on L that is the least distance away from the point given by v, is given by projw v.
(Hint: Construct d(t) which gives the squared-distance of an arbitrary point on L(t) from v.)
Vector Space Axioms
For any vectors u,v,w in a real vector space V , and any real numbers k and l: 1) u + v is in V .
2) u + v = v + u.
3) u + (v + w) = (u + v) + w.
4) There exists a zero vector 0 in V such that u + 0 = u for any vector u.
5) For any u, there exists a vector denoted by −u such that u + (-u) = 0.
6) ku is in V .
7) k(u + v) = ku + kv.
8) (k + l)u = ku + lu.
9) k(lu) = (kl)u.
10) 1u = u.
Page 2
