Calculus Homework Assignment 4
Class:
Student Number:
Name:
1. a. Prove that the midpoint of the line segment from P1 (x1 , y1 , z1 ) to P2 (x2 , y2 , z2 ) is
(
x1 + x2 y1 + y2 z1 + z2
,
,
2
2
2
)
3. a. Draw the vectors a = ⟨3, 2⟩, b = ⟨2, −1⟩,
and c = ⟨7, 1⟩.
b. Show, by means of a sketch, that there are
scalars s and t such that c = sa + tb.
c. Find the exact values of s and t.
[§13.2 #39(a),(b),(d)]
b. Find the lengths of the medians of the triangle with vertices A(1, 2, 3), B(−2, 0, 5), and
C(4, 1, 5).
[§13.1 #19]
2. Find a unit vector that has the same direction as the given vector.
a. ⟨−4, 2, 4⟩
b. 8i − j + 4k
[§13.2 #22, 23]
4. a. Show that the vector ortha b = b−proja b
is orthogonal to a. (It is called an orthogonal
projection of b.)
b. Under what circumstances is
compa b =compb a,
and what circumstances is
proja b =projb a?
[§13.3 #41, 44]
(Over Please)
Calculus Homework Assignment 4
5. Use a scalar projection to show that the distance from a point P1 (x1 , y1 ) to the line ax +
by + c = 0 is
|ax1 + by1 + c|
√
a2 + b2
7.
a. Prove that (a − b) × (a + b) = 2(a × b).
a·c b·c
b. Prove that (a × b) · (c × d) = a · d b · d
[§13.4 #45, 48]
Use this formula to find the distance from the
point (−2, 3) to the line 3x − 4y + 5 = 0.
[§13.3 #49]
6. Let P be a point not on the line L that 8. Suppose a ̸= 0.
passes trough the points Q and R. Show that a. If a · b = a · c, does it follow that b = c?
the distance d from the point P to the line L is b. If a × b = a × c, does it follow that b = c?
c. If a · b = a · c and a × b = a × c, does it follow
|a × b|
d=
that b = c?
[§13.4 #49]
|a|
−−→
−−→
where a = QR and b = QP . Use the formula
to find the distance from the point P (1, 1, 1) to
the line through Q(0, 6, 8) and R(−1, 4, 7).
[§13.4 #43]