Calculus
Homework # 7
Homework # 7
9.3 # 31: (3 pts, p. 653) Find the scalar and vector projections of b onto a.
a = 2i − j + 4k
b = j + 12 k
Scalar Projection: See page 652 for this:
2 · 0 + (−1) · 1 + 4 ·
a·b
Scalar Projection =
= √
|a|
22 + (−1)2 + 42
1
2
√
1
21
=√
=
≈ 0.2182178902
21
21
Vector Projection: Again, see page 652 for this:
Vector Projection =
2 · 0 + (−1) · 1 + 4 · 12
a·b
a
=
(2i − j + 4k) =
|a|2
22 + (−1)2 + 42
2
21 i
−
1
21 j
+
4
21 k
9.4 # 1: (4 pts, p. 661) State whether each expression is meaningful. If not, explain why. If
so, state whether it is a vector or a scalar.
(a) a · (b × c)
(b) a × (b · c)
(c) a × (b × c)
(d) (a · b) × c
(e) (a · b) × (c · d)
(f) (a × b) · (c × d)
Part a: Because both a and the cross product b × c are vectors, the dot product is a
meaningful operation. The result of the dot product will be a scalar value.
Part b: Because the dot product b · c is a scalar value and the cross product operation
requires two vectors, this is not a meaningful operation.
Part c: Because both a and the cross product b × c are vectors, the final cross product is
a meaningful operation. The result of that cross product will be a vector.
Part d: Because the dot product a · b is a scalar value and the cross product operation
requires two vectors, this is not a meaningful operation.
Part e: Because the dot products a · b and c · d are both scalar values and the cross product
operation requires two vectors, this is not a meaningful operation.
Part f: Because the cross products a × b and c × d are both vectors and the dot product
operation requires two vectors, this is a meaningful operation. The result of the dot product
will be a scalar value.
9.5 # 11: (3 pts, p. 671) Is the line through (-4, -6, 1) and (-2, 0, -3) parallel to the line
through (10, 18, 4) and (5, 3, 14)?
While there are several ways to test this, the easiest would be to find the vectors between
each pair of points and then take the cross product. If the value of the cross product is equal
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Calculus
Homework # 7
to zero then the lines are parallel. Note that is does not matter which direction of eacxh of
the two vectors is taken (which end-point is subtracted from the other) or which way the cross
product is taken. That is, if u × v = 0 then v × u = 0. So one version of the vectors and their
cross product are:
u = (−2, 0, −3) − (−4, −6, 1) = 2i + 6j − 4k
v = (5, 3, 14) − (10, 18, 4) = −5i − 15j + 10k


i
j
k


u×v = 2
6
−4 = (60 − 60)i + (20 − 20)j + (−30 + 30)k = 0
−5 −15 10
So these two lines are parallel.
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