ECAT Mathematics
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ECAT Mathematics
Chapter 21: Vectors
82 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
A vector has:
A
Magnitude only
B
Direction only
C
Both magnitude and direction
D
Neither magnitude nor direction
1
A scalar has:
A
Magnitude only
B
Direction only
C
Both magnitude and direction
D
Complex value
2
The magnitude of vector a = (3,4) is:
A
7
B
√7
C
5
D
1
3
If a = (2,3), b = (1,−1): a + b =
A
(3,2)
B
(1,4)
C
(3,2)
D
(2,3)
4
a − b where a=(2,3), b=(1,−1):
A
(1,4)
B
(3,2)
C
(1,2)
D
(2,4)
5
Scalar multiplication: 3a where a=(2,3):
A
(5,6)
B
(6,9)
C
(6,3)
D
(2,9)
6
The zero vector has:
A
Non-zero magnitude
B
Magnitude 0
C
All components 1
D
Undefined direction
7
Unit vector in direction of a=(3,4):
A
(3,4)
B
(3/5,4/5)
C
(1,0)
D
(3/7,4/7)
8
Dot product a·b = ?
A
|a||b|sinθ
B
|a||b|cosθ
C
a₁b₁+a₂b₂+a₃b₃
D
Both B and C
9
If a=(1,2,3) and b=(4,5,6), a·b =
A
4+10+18=32
B
1+2+3=6
C
4+5+6=15
D
12
10
Cross product a×b gives a vector that is:
A
Parallel to both a and b
B
Perpendicular to both a and b
C
Has magnitude a·b
D
Has zero magnitude
11
|a×b| = ?
A
|a||b|cosθ
B
|a||b|sinθ
C
a·b
D
|a|+|b|
12
If a·b=0, vectors a and b are:
A
Parallel
B
Perpendicular
C
Anti-parallel
D
Equal
13
If a×b=0, vectors a and b are:
A
Perpendicular
B
Parallel (or one is zero)
C
Anti-parallel only
D
Different magnitudes
14
Position vector of point P(x,y,z):
A
xi+yj+zk
B
x+y+z
C
(x,y,z) only
D
x²+y²+z²
15
The magnitude of 3i+4j+12k:
A
19
B
13
C
√(9+16+144)=13
D
√(9+16+144)=13
16
i·i = j·j = k·k =
A
0
B
1
C
−1
D
i
17
i·j = j·k = k·i =
A
1
B
−1
C
i
D
0
18
i×j =
A
k
B
−k
C
i
D
j
19
j×i =
A
k
B
−k
C
i
D
j
20
j×k =
A
i
B
−i
C
j
D
k
21
k×i =
A
i
B
j
C
k
D
−j
22
The scalar triple product a·(b×c) represents:
A
Area of triangle
B
Volume of parallelepiped
C
|a×b|
D
a·b+b·c
23
If a·(b×c)=0, the vectors are:
A
Parallel
B
Perpendicular
C
Coplanar
D
Orthogonal
24
The vector projection of a on b:
A
(a·b)/|b|
B
(a·b)b/|b|²
C
a×b/|b|²
D
|a|cosθ
25
Angle between a=(1,0,0) and b=(0,1,0):
A
45°
B
60°
C
90°
D
180°
26
a=(2,3), b=(6,9): these vectors are:
A
Perpendicular
B
Parallel (b=3a)
C
Anti-parallel
D
Unrelated
27
Direction cosines of vector a=(l,m,n): cos α=
A
l
B
l/|a|
C
m/|a|
D
n/|a|
28
Sum of squares of direction cosines l²+m²+n² (after dividing by |a|²):
A
0
B
|a|²
C
1
D
2
29
Vector equation of line through point A with direction d:
A
r=A+td
B
r=A×d
C
r=A+d
D
r=td
30
The cross product is:
A
Commutative
B
Anti-commutative (a×b=−b×a)
C
Associative
D
Distributive only
31
The dot product is:
A
Anti-commutative
B
Non-commutative
C
Commutative (a·b=b·a)
D
Zero always
32
If a=(1,2,3), |a|=
A
6
B
√6
C
√14
D
14
33
Unit vector along i:
A
(1,0,0)
B
(0,1,0)
C
(0,0,1)
D
(1,1,1)
34
Unit vector along j:
A
(1,0,0)
B
(0,1,0)
C
(0,0,1)
D
(1,1,1)
35
Unit vector along k:
A
(1,0,0)
B
(0,1,0)
C
(0,0,1)
D
(1,1,1)
36
Area of parallelogram with adjacent sides a and b:
A
a·b
B
|a×b|
C
|a||b|
D
|a|+|b|
37
Area of triangle with sides a and b:
A
a·b
B
|a×b|
C
(1/2)|a×b|
D
|a||b|sinθ
38
If |a|=3, |b|=4, angle=60°: a·b=
A
12
B
6
C
12cos60°=6
D
Both B and C
39
If |a|=3, |b|=4, angle=90°: |a×b|=
A
12
B
0
C
6
D
12sin90°=12
40
The vector (0,0,0) is:
A
Unit vector
B
Zero vector
C
Position vector
D
Normal vector
41
Resolving vector F=5N at 30° to horizontal: horizontal component=
A
5sin30°=2.5
B
5cos30°=5√3/2
C
5tan30°
D
5
42
Vertical component of F=5N at 30°:
A
5cos30°
B
5sin30°=2.5
C
5tan30°
D
5
43
If three vectors form a closed triangle: a+b+c=
A
1
B
a·b
C
0
D
abc
44
Resultant of forces 3N east and 4N north:
A
7N
B
5N NE
C
5N
D
1N
45
Direction of that resultant (angle from east):
A
53°N of E (arctan(4/3)≈53°)
B
30°
C
45°
D
37°
46
The negative of vector a:
A
Has same magnitude, opposite direction
B
Has zero magnitude
C
Is zero vector
D
Is perpendicular to a
47
Collinear vectors a and b satisfy a=kb for some scalar:
A
Complex k
B
k=0 only
C
Non-zero k
D
k=1 only
48
The Work done by force F on displacement d:
A
F×d
B
|F×d|
C
F·d
D
|F||d|
49
Moment (torque) of force F about a point: M=r×F. |M|=
A
r·F
B
|r||F|cosθ
C
|r||F|sinθ
D
r+F
50
Vector addition is:
A
Only commutative
B
Only associative
C
Both commutative and associative
D
Neither
51
Scalar triple product [a,b,c] = a·(b×c) = b·(c×a) = c·(a×b) shows:
A
Commutativity
B
Cyclic property
C
Anti-symmetry
D
None
52
If a=(1,1,0), b=(1,−1,0): a·b=
A
0
B
2
C
−1
D
1
53
So a and b are:
A
Parallel
B
Perpendicular
C
Anti-parallel
D
Equal
54
The position vector from A(1,2,3) to B(4,6,8):
A
(3,4,5)
B
(5,8,11)
C
(−3,−4,−5)
D
(4,6,8)
55
|AB| where A=(1,2,3) and B=(4,6,8):
A
√(9+16+25)=5√2
B
√50=5√2
C
15
D
√50
56
The midpoint M of AB (A=(1,2,3), B=(4,6,8)):
A
(5/2,4,11/2)
B
(3,4,5)
C
(2,4,5)
D
(5,8,11)
57
Cross product of parallel vectors:
A
|a||b|
B
1
C
0 (zero vector)
D
a+b
58
a×a = ?
A
|a|²
B
2a
C
0 (zero vector)
D
a²
59
The angle between a+b and a−b if a and b are unit vectors and a⊥b:
A
0°
B
90°
C
45°
D
60°
60
For coplanar vectors a,b,c: scalar triple product equals:
A
1
B
0
C
|a||b||c|
D
|a×b|·|c|
61
A vector of magnitude 5 in direction (3,4,0) (|d|=5):
A
(3,4,0) itself
B
(3/5,4/5,0)×5=(3,4,0)
C
(15,20,0)/5
D
(3,4,0)
62
Two vectors are equal if they have same:
A
Magnitude only
B
Direction only
C
Magnitude and direction
D
Starting point
63
Free vector can be:
A
Placed only at origin
B
Placed only at a fixed point
C
Placed anywhere (not tied to a specific point)
D
Has no length
64
In 2D: a=(a₁,a₂) and b=(b₁,b₂). a×b (as scalar/k-component):
A
a₁b₂−a₂b₁
B
a₁b₁+a₂b₂
C
a₁b₂+a₂b₁
D
a₁a₂−b₁b₂
65
For a=(3,1), b=(1,3): a×b (2D)=
A
9−1=8
B
3−3=0
C
8
D
−8
66
The vector 2i+3j−k has components:
A
(2,3,1)
B
(2,3,−1)
C
(3,2,−1)
D
(2,3,0)
67
|2i+3j−k|=
A
6
B
√14
C
√(4+9+1)=√14
D
14
68
The sum of i+j+k:
A
Has magnitude √3
B
Has magnitude 1
C
Has magnitude 3
D
Is zero
69
If a=2i+3j, the direction angle with x-axis:
A
arctan(3/2)
B
arctan(2/3)
C
arcsin(3)
D
60°
70
Orthogonal vectors: a·b=0. Example from {i,j,k}: i·k=
A
1
B
0
C
i
D
k
71
The scalar projection of a on b:
A
a×b/|b|
B
a·b/|b|
C
|a|×|b|
D
a·b×|b|
72
For a=3i+4j (|a|=5), projection on i (x-axis):
A
4
B
3
C
5
D
3/5
73
The vector product i×(j×k) = i×i = ?
A
j
B
k
C
0
D
−k
74
Lagrange's identity: |a×b|²=|a|²|b|²−(a·b)². If a⊥b: |a×b|=
A
0
B
|a|²|b|²
C
|a||b|
D
|a|+|b|
75
The equation of plane through origin with normal n=(a,b,c): ax+by+cz=
A
1
B
n
C
0
D
a+b+c
76
Distance from origin to plane ax+by+cz+d=0:
A
|a+b+c+d|
B
|d|/√(a²+b²+c²)
C
d
D
|d|/|abc|
77
A vector in 3D has how many components?
A
2
B
3
C
4
D
1
78
The cross product is defined for vectors in:
A
2D only
B
3D (and generalizations)
C
All dimensions equally
D
Only unit vectors
79
If F=2i−3j+k and d=i+j+k: Work=F·d=
A
2−3+1=0
B
2+3+1=6
C
−1
D
0
80
i+j+k as unit vector: need to divide by √3. Unit vector=
A
(1/√3)(i+j+k)
B
i+j+k
C
√3(i+j+k)
D
(i+j+k)/3
81
The angle between i and i+j (cos formula):
A
45°
B
60°
C
30°
D
90°
0
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