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QUESTION BANK mathematics (VECTOR ALGEBRA)
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THIS PDF CONTAINS QUESTION BANK OF CLASS 12TH CHAPTER VECTOR ALGEBRA ALONG WITH THE ANSWER KEY AT THE END
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Vector AlgebraQUICK RECAP
VECTOR 8 Magnitude : The distance between the
points A and B is called the magnitude of the
8 A physical quantity having magnitude as well
as direction is called a vector. A vector is directed line segment AB . It is denoted by
represented by a line segment, denoted as | AB | .
AB or a . Here, point A is the initial point 8 Position Vector : Let P be any point in space,
having coordinates (x, y, z) with respect to
and B is the terminal point of the vector AB .
some fixed point O (0, 0, 0) as origin, then
,
they have equal magnitudes and direction
the vector OP having O as its initial point
regardless of the positions of their initial
and P as its terminal point is called the
of
the point P with respect
position vector points.
to
O. The vector OP is usually denoted by r . 8 Coinitial Vectors : Vectors having same
initial point are called co-initial vectors.
8 Collinear Vectors : Two or more vectors are
called collinear if they have same or parallel
supports, irrespective of their magnitudes
and directions.
8 Negative of a Vector : A vector having the
same magnitude as that of a given vector
Magnitude of OP is, OP = x 2 + y 2 + z 2 but directed in the opposite sense
is called
i.e., | r | = x 2 + y 2 + z 2 . negative of the given vector i.e., BA = − AB .
In general, the position vectors of points A, ADDITION OF VECTORS
B, C, etc. with
respect
to the origin O are
denoted by a, b, c , etc. respectively. 8 Triangle law : Let the
8 Direction Cosines and Direction Ratios : vectors be a and b
The angles a, b, g made by the vector r so positioned such
with the positive directions of x, y and z-axes that initial point of
respectively are called its direction angles. one coincides with
The cosine values of these angles, i.e., cosa, terminal point of the
cosb and cosg are called direction cosines of other. If a = AB, b = BC. Then the vector
the vector r , and usually denoted by l, m and
n respectively. a + b is represented
by the
side of DABC
third
Direction cosines of r are given as i.e., AB + BC = AC
x y 8 Parallelogram law :
l= ,m = ,
x2 + y2 + z2 x2 + y2 + z2 If the two vectors a
z and b are represented
n= by the two adjacent
x + y2 + z2
2
sides OA and OB
The numbers lr, mr and nr, proportional
to of a parallelogram
the direction cosines of vector r are called
OACB, then their sum a + b is represented
direction ratios of the vector r and denoted
in magnitude and direction by the diagonal
as a, b and c respectively.
i.e., a = lr, b = mr and c = nr OC of parallelogram
OACB
through
their
Note : l2 + m2 + n2 = 1 and a2 + b2 + c2 ≠ 1, common point O i.e., OA + OB = OC
(in general). Properties of Vector Addition
TYPES OF VECTORS X Vector is commutative i.e.,
addition
a + b = b + a.
8 Zero vector : A vector whose initial and
terminal points coincide is called a zero (or X Vector
addition i.e.,
isassociative
null) vector. It cannot be assigned a definite a + (b + c) = (a + b) + c.
direction as it has
zero magnitude and it is X Existence of additive identity : The zero
denoted by the 0 . vector
acts identity i.e.,
as additive
8 Unit Vector : A vector whose magnitude is a + 0 = a = 0 + a for any vector a .
unity i.e., | a | = 1 . It is denoted by a . X Existence
of additive inverse : The negative
of a i.e., − a acts as additive inverse i.e.,
8 Equal Vectors : Two vectors a and b
a + (−a) = 0 = (−a) + a for any vector a.
are said to be equal, written as a = b , iff
, MULTIPLICATION OF A VECTOR BY A = ( x2 i + y2 j + z2 k ) − ( x1 i + y1 j + z1 k )
SCALAR
= ( x2 − x1 )i + ( y2 − y1 ) j + ( z2 − z1 )k
8 Let a be a given vector andl be a given
scalar (a real number), then λ a is defined as ∴| P1P2 |= ( x2 − x1 )2 + ( y2 − y1 )2 + (z2 − z1 )2
the multiplication of vector a by the scalar l. SECTION FORMULA
Its
magnitude
is λ times the modulus of
8 Let A, B be two points such that
a i.e., λ a = λ a .
OA = a and OB = b.
Direction of λa is same as that of a if λ > 0 X The position vector r of the point P which
and opposite to that of a if λ < 0. divides the line segment AB internally in the
1
Note : If l = , provided that a ≠ 0, then mb + na
|a | ratio m : n is given by r = .
m+n
λa represents the unit vector in the direction
X The position vector r of the point P which
a
of a i.e. a
= divides the line segment AB externally
in the
|a| mb − na
ratio m : n is given by r = .
COMPONENTS OF A VECTOR m−n
X The position vector r of the mid-point
of the
8 Let O be the origin and P(x, y, z) be any point a+b
in space. Let iˆ , jˆ , kˆ be unit vectors along line segment AB is given by r = .
the X-axis,
Y-axis and Z-axis respectively. 2
Then OP = xiˆ + yjˆ + zk
ˆ , which is called the PRODUCT OF TWO VECTORS
component form of OP .
Here x, y and z are 8 Scalar (or dot) product : The scalar (or dot)
scalar components of OP
and xi , y j , zk are product of two (non-zero) vectors a and b ,
vector components of OP . denoted by a ⋅ b (read as a dot b ), is defined
8 If a and b are two given vectors as as a ⋅ b = a b cos θ = ab cos θ,
a = a iˆ + a jˆ + a kˆ and b = b iˆ + b jˆ + b kˆ where, a = a , b = b and
1 2 3 1 2 3 q(0 ≤ q ≤ p) is the
and l be any scalar, then angle between a and b.
X a + b = (a1 + b1 )
i + (a2 + b2 )
j + (a3 + b3 )k
X Properties of Scalar Product :
X a − b = (a1 − b1 )
i + (a2 − b2 )
j + (a3 − b3 )k
(i) Scalar product is commutative : a ⋅b = b ⋅ a
λa = ( λa1 )
i + ( λa2 )
j + ( λa3 )k
X (ii) a ⋅ 0 = 0
X a =b ⇔
a1 = b1 , a2 = b2 and a3 = b3 (iii) Scalar product is distributive over
X a and b are collinear iff addition :
b1 b2 b3
= = = λ. • a ⋅ (b + c ) = a ⋅ b + a ⋅ c
a1 a2 a3
• (a + b ) ⋅ c = a ⋅ c + b ⋅ c
VECTOR JOINING TWO POINTS
(iv) λ(a ⋅ b ) = (λ a ) ⋅ b = a ⋅ (λ b ), λ be any
8 If P1(x1, y1, z1) and scalar.
P2 (x2, y2, z2) are (v) If i, j, k are three unit vectors along
any two points in three mutually perpendicular lines, then
the space then the i ⋅ i = j ⋅ j = k ⋅ k =1 and i ⋅ j = j ⋅ k = k ⋅ i = 0
vector joining P1 (vi) Angle between two non-zero vectors
and P2 is the
a ⋅b
vector P1P2 . a and b is given by cosθ =
a b
Applying
law in DOP1P2, we get
triangle
a ⋅b
OP1 + P1P2 = OP2
i.e., q = cos −1
⇒ P1P2 = OP2 − OP1 | a || b |
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