082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

256 Chapter 7 Vectors
16 Show that the vectorsi −j and −3i −3j are
perpendicular.
17 Find the norm ofeach ofthe vectors
⎛ ⎞ ⎛ ⎞
7 2
⎜ 2
⎟
⎝ −1⎠ and ⎜ 1
⎟
⎝ 0 ⎠
2 −4
18 (a)Usethe scalar product to findthe value ofthe
scalar µ sothat i +j+µk isperpendicular to the
vector i +j+k.
(b)Usethevectorproductandtheresultsfrompart(a)
to findamutually perpendicular setofunitvectors
ˆv 1 , ˆv 2 and ˆv 3 , where ˆv 1 is inclined equally to the
vectorsi,jand k.
19 ThepointsA, Band C have coordinates (2,−1, −2),
(4,−1,−3)and (1,3,−1).
(a) Writedown the vectors → ABand → AC.
(b) Using the vectorproduct findaunitvector
which isperpendicular to the plane containing
A, B and C.
(c) IfDisthe point with coordinates (3,0,1),use
the scalar product to find the perpendicular
distance fromDto the plane ABC.
20 Thecondition forvectorsa,band cto be coplanar
(i.e.they liein the same plane) isa · (b ×c) = 0.
(a) Show that the vectorsa = 4i +5j +6k,
b=6i−3j−3kandc=−i+2j+2kare
notcoplanar.
(b) Givend = −i +2j + λk,findthe value of λ so
thata, b anddare coplanar.
21 Points Aand Bhave position vectorsaand b
respectively. Show that the position vector ofan
arbitrarypoint on the line AB isgiven by
r = a + λ (b −a) forsome scalar λ.Thisis the
vector equationofthe line.
22 Usevector methods to show that the threemedians of
any triangle intersect atacommon point (called the
centroid).
23 Usethe vectorproduct to findthe area ofatriangle
with vertices atthe pointswith coordinates (1,2,3),
(4,−3,2),and (8,1,1).
Solutions
1 (a) 24, −7i −32j +17k (b) 18, 0
2 0
3 √ 54, √ 66,57,0.9548
4 √ 41, √ 19, √ 154,0.4446
5
1
√
139
a,
1
√
20
b,
6 5i+7j−4k,22
1
√
2296
(−12i −6j −46k)
7 (a×b)×c=4i−12j−16k
a·c=8,b·c=4
a×(b×c)=−2i−22j−8k
1 1
9 √ (i +7j), √ (i +2j),18.4 ◦
50 5
10
µq
56 √ (i +11j +7k)
14π
12 (a) 3i−5j−4k
(b) 5j −2k
13 √ 24 = 4.90
14 87.5 ◦
15
1
√
230
(5i +14j −3k)
17 √ 58, √ 21
18 (a) µ=−2
(b) ˆv 1 = 1 √
3
(1,1,1), ˆv 2 = 1 √
6
(1,1,−2),
ˆv 3 = 1 √
2
(−1,1,0)
19 (a) (2,0,−1), (−1,4,1) (b)
1
(4,−1,8) (c) 3
9
20 (a) a · (b ×c) = 9 ≠ 0 andhence the vectors are not
coplanar
(b) λ = 31/14
23 1 2√
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