Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
VECTORS
• Position vector of a point P (x, y, z) in space is given by
= = x + y + z
, and its
• Magnitude is given by |
| =
• The scalar components (x, y, z) of a vector are its direction ratios, and represent its projections on the
respective axes.
• Direction cosines (l, m, n) are the cosine of the angles (α, β, γ) the vector make with positive direction
of the respective axes.
• The magnitude r and direction ratios x, y, z and direction cosines l, m, n are related as:
o l = cos α = x/r; m = cos β = y/r, n = cos γ = z/r, ands
o
Sum of two vectors:
• The vector sum of 3 sides of a triangle taken in order is
.
• The vector sum of two coinitial vectors is given by the diagonal of the parallelogram whose adjacent
sides are the given vectors.
Multiplication of a vector by a scalar:
• The multiplication of a given vector by a scalar k changes it magnitude by a multiple of |k|:
o If k is +, then direction remains same,
o If k is - direction reverses.
• Unit vector in the direction of a vector is given by: /|
• Vector of magnitude m in the direction of
is given by: m
/|
Section Formula:
• The PV of a point P dividing the line segment joining the point A and B whose PV’s are and
respectively, in the ration m:n is given by:
o m
+ n / m + n (for internal division)
o m
- n / m - n (for external division)
Scalar (dot) Product of two vectors:
• For two vectors and
having an angle θ between them is defined as:
.
= |||
| cos θ.
• When dot product is known, angle b/w two vectors will be:
cos θ = .
/ |||
|
• Properties of dot product:
o Dot product is scalar quantity.
o Dot product is commutative. i.e. .
=
.
o . = .
=
. = 0 (since angle is 90
o
and cos 90
o
= 0)
o . = . =
.
= 1 (since angle is 0
o
and cos 0
o
= 1)
Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
Vector (cross) Product of two vectors:
• For two vectors and
having an angle θ between them is defined as:
x
= |||
| sin θ ,
where,
is a unit vector perpendicular to the plan containing and
given
by right handed system of co-ordinate axis.
• Properties of cross product:
o Cross product is a vector quantity.
o Cross product is not commutative. i.e. x
= -
x
o x =
; x
= ;
x = (since angle is 90 and sin 90 = 1, so we get the 3
rd
vector using right
hand rule)
o Similarly, x = -
;
x = - ; x
= -
o x = x =
x
= 0 (since angle is 0
o
and sin 0
o
= 0)
Vector operations in component form:
o Sum:
(a
1
+ a
2
+ a
3
) + (b
1
+ b
2
+ b
3
) = (a
1 +
b
1
)
+ (a
2
+ b
2
) + (a
3
+ b
3
)
o Scalar multiplication:
(a
1
+ a
2
+ a
3
) a
1
+ a
2
+ a
3
o Dot product:
.
= a
1
b
1
+ a
2
b
2
+ a
3
b
3
o Cross product:
x
=
