Chapter 1 Exercise 1A
Q. 1.
(i)
Q. 4.
Q
→
→
→
x
b
a+
a
1→ 1→
–x + –y
2
2
→
→
y
→
b
Q. 5.
Q
→
a
(ii)
→
→
a+
→
a
→
b
→
c
→
a+b
(a→+ →
b) + →
c
→
b
→
b
(iii)
→
a
Q. 6.
Q
→
a+
→
→
→
b
a
b
→
a + (→ →
b+c
)
→
b
b→
+ →
c
→
c
→
(iv)
→
a+
a
b
→
__›
___›
__›
__›
___›
__›
Yes; (a + b ) + c = a + (b + c ).
→
b
(v)
→
Q. 7.
Q
→
b
→
a+b
→
a
→
y
___›
→
→
3x + 1.5y
(vi) 0 , The null vector
→
x
Q. 2.
→
→
2a
+b
→
a
Q. 8.
Q
→
→
x
→
2
Q. 3.
x→–
y
–1 →
b
→
y
y
→
x
1→
→ – –2x
y
→
–
4x
→
y
Q. 9.
Q
5 cm; E 53° N.
1
Q. 10.
Approximately 7 cm due East.
Q. 11.
13 cm.
_______
Q
Q. 6.
(i) Magnitude = √ 32 + 42 = 5
__
_›
1 ›
∴ Unit vector = __(3i + 4j )
5
__
__›
1__ ›
(ii) ___
(i + 2 j )
√5
__
__
1__ › ›
(iii) ___
(i − j )
√2
__›
__›
1
___
(−3i − j )
(iv) ____
√10
__
__
1 __ › ›
__
(v) (√ 3 i + j )
2
Q
Q. 7.
k(2i − j ) + l(4i + 3 j ) = 2i − 11j
___
(i) √ 29 , E 21° 48’ N.
__
(ii) √ 8 , NE.
(iii) 5, E 36° 52’ S.
(iv) 13, W 67° 23’ S.
___
(v) √ 20 , W 26° 34’ N.
__
(vi) √ 2 , NE.
1__
, SE.
(vii) ___
√2
(viii) 1, W 53° 8’ S.
(x) √ 12 , E 30° N.
Q. 8.
Q
__›
__›
__›
__›
__›
___›
__›
= 5i − 4j
__›
__›
__›
__›
__›
__›
__›
__›
__›
__›
= −i − 2j
__›
__›
___›
__›
__›
__›
__›
__›
__›
__›
__›
Q.
Q 11.
__›
__›
= 12i + 5j
∴ k = ±2
Q.
Q 12.
(i) √ 13
____
(ii) √ 104
________
__›
(iii) |x + y | = √ 122 + 52 = 13
__›
__›
___
= 3.606 + 10.20
__›
= 13.806
__›
__›
__›
−2
___
________
___
___
___
(iii) √ 80 < √10 + √ 50
since 8.944 < 10.233
20 −40
× ___ = ____ = −1
8
40
5
Q.
Q 16.
−2 −6
× ___ = ___ = −1
6
6
1
∴⊥
3
__
−t
___
9
6
× __ = −1
2
∴t=3
__›
(ii) √ 16 + 64 = √80 = 8.944
2
Q
Q. 15.
(i) 4i + 8 j
___
−8 −24
× ___ = ____ = −1
4
24
6
∴⊥
3
__
∴⊥
__›
∴|x + y | < |x | + |y |
(since 13 < 13.806)
Q. 4.
Q. 14.
Q
____
(iv) |x | + |y | = √ 13 + √104
___
√2k2 = √50
∴ k = ±5
Q
Q. 13.
___
____
√121 + k2 = √125
____
x + y = (2i + 3j ) + (10i + 2j )
__›
____
________
= 7j
__›
_______
∴ p = ±5
__›
__›
__›
√50 = √2p2
(iv) 2a − 3b = 2(3i − j ) − 3(2i − 3j )
Q. 3.
__›
√65 = √49 + k2
___
Q
Q. 10.
(iii) b − a = (2i − 3j ) − (3i − j )
__›
__›
∴ k = ±4
__›
= i + 2j
__›
__›
__›
__›
__›
4i − 2 j + t(7i + 5 j ) = ki + 0 j
___
Q
Q. 9.
(ii) a − b = (3i − j ) − (2i − 3j )
___›
__›
∴ t = 0.4
(i) a + b = (3i − j ) + (2i − 3j )
__›
__›
∴ 4 + 7t = k and −2 + 5t = 0
(xi) 4, due West.
___›
__›
Solving gives l = −2, k = 5
___
__›
__›
∴2k + 4l = 2 and −k + 3l = −11
(ix) 2, W 30° N.
Q. 2.
__
√5 ≥ √20 − √5
since 2.236 ≥ 4.472 − 2.236
Exercise 1B
Q. 1.
___
__
Q
Q. 5.
Q. 17.
Q
p
__
−2
× ______ = −1
p+1
∴ 4p + 4 = 2p
4
∴ p = –2
Q
Q. 5.
x
Q. 1.
(i) |AB| = H cos q; |BC| = H sin q
(ii) |AB| = H sin q; |BC| = H cos q
(iii) |AC| = H cos q; |BC| = H sin q
5
12
(i) cos A = ___, sin q = ___
13
13
35
12
(ii) cos A = ___, sin q = ___
37
37
__
__
√7
√7
___
___
(iii) sin A =
, tan A =
4
3
(iv) cos =
( )( )
7 ___
1__
7
___
∴ x = H ____
= ___ H
10
√50 √2
∴ H : x = 10 : 7
Exercise 1D
E
40
___
Q. 1.
Q
41
__›
__›
__›
(i) |AB| = 4√ 3 cm; |BC| = 4 cm
__ __›
__›
__›
__›
= 9.511i + 3.09j
(ii) |XY| = 2 m; |YZ| = 2 m
__›
__›
(iii) 8 cos 45° i − 8 sin 45° j
__ __›
(iii) |AB| = 10 sin 40° = 6.428 m
__ __›
= 4√ 2 i − 4√ 2 j
|BC| = 10 cos 40° = 7.66 m
__›
__›
(iv) − 20 cos 20° i + 20 sin 20° j
__›
(iv) |XY| = 20 cos 35° = 16.38 cm
__›
= −18.794i + 6.84 j
|XZ| = 20 sin 35° = 11.47 cm
__›
___
__›
___
(v) −√ 50 cos 45° i − √ 50 sin 45° j
__›
(v) |PQ| = 40 cos 20° = 37.59 m
__›
= −5 i − 5 j
|QR| = 40 sin 20° = 13.68 m
__›
__›
(vi) 12 cos 39° i − 12 sin 39° j
__›
(vi) |PQ| = 12 cos 60° = 6 m
__›
= 9.3252 i − 7.5516 j
__
|RQ| = 12 sin 60° = 6√ 3 m
(vii) |AB| = 15 cos q = 12 cm
__›
(i) 2 cos 60° i + 2 sin 60° j = i + √ 3 j
(ii) 10 cos 18° i + 10 sin 18° j
__
Q. 3.
H
x = ADJ = H cos a cos b
(iv) |AC| = H cos q; |AB| = H sin q
Q. 2.
b α
H cos a
Exercise 1C
Q.
Q 2.
|BC| = 15 sin q = 9 cm
11
(viii) |RQ| = 78 cos a = 30 m
|PQ| = 78 sin a = 72 m
A
___
(ix) |XY| = √ 13 cos q = 3
4
___
|YZ| = √ 13 sin q = 2
___
(x) |AB| = √20 cos a = 4
___
|BC| = √ 20 sin a = 2
__›
___›
__›
u = −10 cos a i − 10 sin a j
__›
_›
__›
= −8i − 6 j
__›
__›
v = 13 cos b i − 13 sin b j
H sin q
Q. 4.
H
q
___›
f
x
x = ADJ = H sin q cos f
3 5
∴ x = (13) __ ___ = 3
5 13
( )( )
__›
_›
= 12i − 5j
__›
_›
__›
∴ u + v = 4i − 11j
___›
___›
__›
w = −(u + v )
_›
__›
= −4i + 11j
3
____________
___›
|w | = √ (−4)2 + (11)2
Exercise 1E
E
____
= √ 137
Q. 1.
Q
= 11.7
_›
⇒ 10 − 26 cos a = 0
5
⇒ cos a = ___
13
12
⇒ sin a = ___
13
__›
__
p = √ 8 cos 45° i + √ 8 sin 45° j
__›
_›
_›
___›
q = 4 cos 30° i − 4 sin 30° j
___
= 2√ 3 i − 2 j
__›
r = −10 cos 40° i − 10 sin 40° j
__›
= −7.66i − 6.428j
Q
Q. 2.
__›
_›
__›
Q. 5.
No i -component
⇒ 4 − 5 cos a = 0
4
⇒ cos a = __
5
3
⇒ sin a = __
5
_›
___›
__›
__›
b = −13 cos a i + 13 sin a j
__›
_›
= −12i + 5 j
___›
__›
___
_›
__›
Q. 6.
___›
Q.
Q 3.
(i) x = −25 cos Ai + 25 sin Aj
__›
_›
__›
_›
_›
_›
___›
__
1 ›
__
_›
_›
(ii) ∴ x + y = −20i + 15j + 8i − 15j
_›
j -component
__›
__›
(iii) |x + y | = 12 units
4
xi +
2
xj
___›
1
__
_›
x)i +
__
_
√3 ›
___
xj
2
_›
1
No i -component ⇒ − 4 + __ x = 0
2
⇒x=8
2
___›
= −12i , which
has no
_
›
2
__
_
√3 ›
___
∴ a + b = (−4 +
_›
__›
__›
__›
y = 17 cos Bi − 17 sin Bj
= 8i − 15j
Let |b | = x
b=
__›
__›
( ))
›
___›
a = −4i
__›
= −20i + 15j
_
= 0i − j
(iii) |a + b | = 5 units
__›
(
›
__›
= 5 j , along the j -axis
__›
_
__›
›
›
3 ›
∴ a + b = 0i + − 4 + 5 __ j
5
_
_
__›
_›
(ii) ∴ a + b = 12i − 12i + 5 j
__›
__›
_›
_›
_›
(i) a = 12i
__›
∴ a + b = (4 − 5 cos a)i + (−4 + 5 sin a)j
∴ r + s + t = −2.6 i + 5.8 j
__›
___›
___›
_›
__›
__›
b = −5 cos ai + 5 sin a j
= 10.3367 i + 3.762 j
__›
__›
__
= 4i − 4j
t = 11 cos 20° i + 11 sin 20° j
_›
( )
__›
___›
_›
_›
__›
( )
_›
__›
_›
_
___
___
__›
1__ ›
1__ ›
a = √ 32 ___
i − √ 32 ___
j
√2
√2
s = −10 cos 58°i + 10 sin 58° j
= −5.299 i + 8.48 j
___›
__›
∴ |a + b | = 24 units
__›
_›
__
= 24 j
∴ p + q = ( 2 + 2√ 3 ) i + 0 j
_›
( )
__›
_›
__ _›
___›
_
__›
›
›
12 ›
∴ a + b = 0i + 26 ___ j
13
_›
__ _›
__›
__›
_›
= 2i + 2 j
Q. 4.
_›
No i -component
__
___›
___›
__›
Direction is W 70° N.
_›
__›
__›
∴ a + b = (10 − 26 cos a)i + 26 sin a j
⇒ A = 70°
Q. 3.
___›
b = − 26 cos a i + 26 sin a j
11
Tan A = ___
4
___›
_›
__›
a = 10i
i.e. |b | = 8
__›
___›
__ _›
_›
∴ a + b = 0i + 4√ 3 j
__›
___›
__
|a + b | = 4√ 3 units
Q. 4.
__›
_›
_›
Q
Q. 6. (i) p = 35 cos a i + 35 sin a j
__
_
3 ›
4 ›
= 35 __ i + 35 __ j
5
5
__›
= 21i + 28j
__›
_›
()
= − 3.420i + 9.397 j
__›
__›
__›
Let |s | = x ⇒ s = x i
_›
_›
__›
__›
_›
__›
___›
a = 10 cos q i + 10 sin q j
()
()
__›
__
___›
___›
( )
___›
_›
_›
_›
_›
_›
_›
33
∴ tan A = ___ = 1
33
⇒ A = 45°
_›
Let |b | = x ⇒ b = −x j
__›
_
_
= 33i + 33 j
= 8i + 6j
___›
q
(ii) ∴ p + q = 21i + 28 j + 12 i + 5 j
3 ›
4 ›
= 10 __ i + 10 __ j
5
5
_›
__›
__›
( )
__›
_
p
A
+
5 ›
12 ›
= 13 ___ i + 13 ___ j
13
13
_›
_›
= 12 i + 5 j
i.e. |s | = 3.420
__›
→
→
q = 13 cos b i + 13 sin b j
No i -component ⇒ x = 3.420
__›
()
___›
∴ r + s = (−3.420 + x)i + 9.397j
Q. 5.
__›
__›
___›
i = − 10 cos 70° i + 10 sin 70° j
_›
_›
∴ a + b = 8i + (6 − x) j
_›
__›
= k i + 0j
∴ x = 6 and k = 8
Q. 7.
___›_
_›
__›
r = − 3.42 i + 9.397j
__›
Let |s | = x
__›
_›
_›
__›
__›
∴ s = x cos 10°i + x sin 10°j
= 0.9848xi + 0.1736x j
__›
__›
__›
__›
∴ r + s = (−3.42 + 0.9848x)i + (9.397 + 0.1736x) j
_›
__›
__›
_›
Since (r + s ) is in a NE direction, the i and j components must be equal.
∴ −3.42 + 0.9848x = 9.397 + 0.1736x
⇒ x = 15.8
Q
Q. 8
8.
____›
_______
___›
__________
___
(i) |p | = √ 82 + 12 = √65
___
|q | = √ (−7)2 + 42 = √65
___›
___›
∴ |p | = |q |
_›
1
__
(ii) “Slope” of p = _›
= = m1
i -bit 8
4
4
“Slope” of q = ___ = − __ = m2
−7
7
1
m1 × m2 = −___ ≠ −1
14
∴ Not perpendicular
__›
_______
___
Q. 9. |−7i + j | = √ 49 + 1 = √ 50
_›
_›
__›
___›_
_›
___›
a = 10 cos 80° i + 10 sin 80° j
__›
_›
= 1.736 i + 9.848 j
___›
Let |b | = x
j -bit
____
_›
Q. 10
10.
Q
___›
__›
Therefore, b = xi
__›
___›
_›
_›
∴ a + b = (1.736 + x)i + 9.848j
___›
___›
Since the direction___ of (a + b ) is 21° 48’,
___›
›
the slope of (a + b ) must equal tan 21° 48’.
9.848
∴ _________ = 0.4
1.736 + x
⇒ x = 22.88
_›
|p(i + j )| = |pi + p j |
_______
____
= √ p2 + p2 = √ 2p2
∴ 2p2 = 50
⇒ p = 5 (sin ce p > 0)
5