2012 vectors (basic) tutorial solutions barely passed
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7/29/2019 2012 Vectors (Basic) Tutorial Solutions Barely Passed
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CJC MATHEMATICS DEPARTMENT2012 JC1 H2 MATHEMATICSTOPIC: VECTORS (BASIC)
Unit Vectors, Parallel Vectors, Collinear Points,Properties of a Parallelogram
1.
=
=
=
4
0
,
2
5
3
,
2
1
1
pOCOBOA
(i)
=
=
4
6
2
2
5
3
2
1
1
BA
Unit vector in the direction ofBA
=
=
2
3
1
14
14
4
6
2
56
1
(ii)
=
4
6
2
BA
=
==
2
1
1
4
0
2
1
1
ppOCOACA
ForA,B, Cto be collinear, CAkBA =
4
)1(26
2n,observatioBy
21
1
46
2
=
=
=
=
p
p
k
pk
(iii) Ifp = 1,
=
4
1
0
OC
=
4
6
2
BA
ForABCD to be a parallelogram,
=
=
=
8
7
2
41
0
46
2
OD
OD
CDBA
(iv) Suppose 3|| =AC 3
2
1
1
=
+
p
1or3
0)1)(3(
032
9215
32)1(1
2
2
22
==
=+
=+
=+++
=+++
pp
pp
pp
pp
p
Scalar Product (Dot Product)
2(a) Length of projection ofa on b = |a ^b |
++
=
8
3
1
831
1
5
4
2
222
=1
74| 2 12 + 40| =
30
74
(b) Length of projection ofa on b = |a ^b |
++
=
4
2
5
425
1
2
7
3
222
=1
45|15 14 8| =
7
45
3
Given
=
=
=
11
1
2
,
7
5
2
,
6
8
2
OROQOP
By Ratio Theorem,
=
3
4
1
OA
=
+
=+=
8
4
2
411
1
2
7
5
2
3
43 OROQOB
(i) Shortest distance fromB to line OP
=|| OP
OPOB
=
6
8
2
104
1
8
4
2
C
A B
D
O P
QR
A
B
C
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7/29/2019 2012 Vectors (Basic) Tutorial Solutions Barely Passed
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13
370
104
2960
24
28
40
104
1==
= (shown)
(ii)
=
=
5
0
3
8
4
2
3
4
1
BA
=
=
3
3
0
8
4
2
11
1
2
BR
25999
5
0
3
3
3
0
||||cos
++
=
=
BABR
BABR
0
3.127=
Vector Product (Cross Product)
4.
AB =
OB
OA =
32
1
01
2=
31
1
AC=
OC
OA =
1
1
0
0
1
2
=
1
2
2
n =
31
1
12
2=
45
7
Unit vector =1
42+5
2+7
2
45
7=
1
90
45
7
5. ab = ac
(ab) (ac) = 0 a (b c) = 0
a is parallel to (b c)Hence b c = ka, where kis a scalar (shown)
6. [10/NJC/I/3]
(i) ( )AB OP = b auuuv uuuv
p
= b p a p
= a p a p (since b p = a p)= 0
Hence,AB is perpendicular to OP.
OR
b p = a pb p a p = 0
( )b a p = 0
0AB OP =uuuv uuuv
Hence,AB is perpendicular to OP.
(ii) Since =a b , then P must be the midpoint ofAB.
Using ratio theorem, ( )12
OP = +a buuur
Thus, 2OD OP=uuuv uuuv
( )1
22
= +
a b = a + b
(iii)
a b represents the
area of rhombus OADB or OBDAOR
magnitude of a vector which is perpendicular to
a and b
A B
O
D
P