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Spring 2015
Math 152
Quiz 9 Solutions Wed, 22/Apr/2015
1. Find the two unit vectors that are parallel to the
tangent line to the parabola y = x2 at the point (2, 4).
• The slope of the tangent line at the point is
dy 4
rise
dx x=2 = 2x x=2 = 4 = 1 = run .
• A vector parallel to the tangent line is v = [1, 4].
• Two unit vectors parallel
to thei tangent line are
h
[1,4]
v
1
u = kvk = √1+16 = √17 , √417 and
h
i
−u = − √117 , − √417 .
2. Find an equation of the sphere with center C (2, −6, 4)
and radius r = 5. Then describe its intersection with
each of the coordinate planes x = 0 (yz-plane), y = 0
(xz-plane), z = 0 (xy-plane), both geometrically (in
English) and analytically (equations). [Some of the
intersections may be the empty set, 0.]
/
• The sphere has equation
(x − 2)2 + (y + 6)2 + (z − 4)2 = 25.
• When x = 0 is substituted into the sphere’s
equation, we have (y + 6)2 + (z − 4)2 = 21,
x = 0. Hence the sphere intersects the yz-plane
in the
√ circle with center C (0, −6, 4) and radius
r = 21.
• When y = 0 is substituted into the sphere’s
equation, we have (x − 2)2 + (z − 4)2 = −11,
a contradiction, since a sum of squares of real
numbers is never negative. Hence the sphere
does not intersect the xz-plane. In other words,
the intersection is the empty set 0.
/
• When z = 0 is substituted into the sphere’s
equation, we have(x − 2)2 + (y + 6)2 = 9, z = 0.
Hence the sphere intersects the xy-plane in the
circle with center C (2, −6, 0) and radius r = 3.
3. Find the three angles of the triangle with vertices
A (1, 0, −1), B (3, −2, 0), C (1, 3, 3), in degrees.
• Label the angles at vertices A, B, C as α, β , γ,
respectively. Manipulate the Physics definition
of dot product:
v·w
.
v · w = kvk kwk cos θ =⇒ θ = cos−1 kvkkwk
!
• So α = cos−1
−
→−
→
AB·
−
AC
→
→−
ABAC
≈ 97.66◦ . Similarly,
β ≈ 53.50◦ and γ ≈ 28.84◦ ; α + β + γ = 180◦ .
4. Find the work done by a force F = 8i − 6j + 9k that
moves an object from the point P (0, 10, 8) to the point
Q (6, 12, 20) along a straight line. Here distance is
measured in meters and the force in newtons. Include
correct units of work.
• The displacement vector is
D = Q − P = [6, 2, 12].
• The work done is
W = F · D = 48 − 12 + 108 = 144 joules.
5. Use vectors to determine whether the triangle with
vertices P (1, −3, −2), Q (2, 0, −4), and R (6, −2, −5)
is right-angled.
• Form three vectors which represent the sides of
the triangle.
u = Q − P = [1, 3, −2]
v = R − Q = [4, −2, −1]
w = P − R = [−5, −1, 3]
• Compute the lengths of these vectors.
√
√
√
kuk = 14, kvk = 21, kwk = 35
• Since 14 + 21 = 35, these lengths satisfy the
Pythagorean Theorem. Hence the triangle is
right-angled!
6. [BONUS] Find values ofx such that the
angle
2
1
−1
between
the
vectors
v
=
and
◦
w = 1 x 0 is 45 . Give exact values
for x and approximations to 2 decimal places.
• Start with the Physics definition of dot product
and see where the road leads.
v · w = kvk kwk cos 45◦
√ p
1
2+x =
6
1 + x2 √
2
2
2
4 + 4x + x = 3 1 + x
0 = 2x2 − 4x − 1
√
4 ± 16 + 8
x =
4
√
1
x = 1± 2 6
x ≈ −0.22, 2.22
• “All roads lead to Rome!”