Cheat Sheet: Vectors | Physics Class 11 - NEET PDF Download

 Page 1


Vectors Cheat Sheet
Introduction
Vectors is a fundamental chapter in Physics, essential for understanding quantities with
both magnitude and direction. It forms the basis for analyzing motion, forces, and
other physical phenomena in multiple dimensions, making it crucial for solving complex
problems in various physics exams.
Scalars and Vectors
Type Description Examples
Scalar Quantities with only magnitude,
no direction.
Mass, time, speed, energy
Vector Quantities with both magnitude
and direction.
Displacement, velocity,
force, acceleration
Representation of Vectors
• Notation: Vectors are denoted by bold letters (e.g., A) or with an arrow (e.g.,
- ?
A).
• Components: A vector in 2D: A = A
x
i+A
y
j, where i, j are unit vectors along x
and y axes.
• Magnitude: |A| =
v
A
2
x
+A
2
y
for a 2D vector.
• Direction: ? = tan
-1
(A
y
/A
x
), measured from the positive x-axis.
Types of Vectors
Type Description
Unit Vector Vector with magnitude 1, e.g., ˆ a =A/|A|.
Zero Vector Vector with zero magnitude, no speci?c direction.
Equal Vectors Same magnitude and direction, e.g., A =B.
Parallel Vectors Same or opposite direction, di?erent magnitudes.
1
Page 2


Vectors Cheat Sheet
Introduction
Vectors is a fundamental chapter in Physics, essential for understanding quantities with
both magnitude and direction. It forms the basis for analyzing motion, forces, and
other physical phenomena in multiple dimensions, making it crucial for solving complex
problems in various physics exams.
Scalars and Vectors
Type Description Examples
Scalar Quantities with only magnitude,
no direction.
Mass, time, speed, energy
Vector Quantities with both magnitude
and direction.
Displacement, velocity,
force, acceleration
Representation of Vectors
• Notation: Vectors are denoted by bold letters (e.g., A) or with an arrow (e.g.,
- ?
A).
• Components: A vector in 2D: A = A
x
i+A
y
j, where i, j are unit vectors along x
and y axes.
• Magnitude: |A| =
v
A
2
x
+A
2
y
for a 2D vector.
• Direction: ? = tan
-1
(A
y
/A
x
), measured from the positive x-axis.
Types of Vectors
Type Description
Unit Vector Vector with magnitude 1, e.g., ˆ a =A/|A|.
Zero Vector Vector with zero magnitude, no speci?c direction.
Equal Vectors Same magnitude and direction, e.g., A =B.
Parallel Vectors Same or opposite direction, di?erent magnitudes.
1
Vector Addition
• Parallelogram Law: If A and B are adjacent sides of a parallelogram, resultant
R is the diagonal.
– Magnitude: |R| =
v
A
2
+B
2
+2ABcos?, where ? is the angle between A
and B.
– Direction: tan? = (Bsin?)/(A+Bcos?).
• Triangle Law: Place tail of B at head of A; R is from tail of A to head of B.
• Component Method: R =A+B = (A
x
+B
x
)i+(A
y
+B
y
)j.
Vector Subtraction
• A-B =A+(-B), where-B is B reversed in direction.
• Magnitude and direction calculated similarly to addition.
Dot Product (Scalar Product)
• De?nition: A·B = ABcos?, where ? is the angle between A and B.
• Properties:
– Result is a scalar.
– A·B = A
x
B
x
+A
y
B
y
(in component form).
– A·A = A
2
; i·i =j·j = 1; i·j = 0.
Cross Product (Vector Product)
• De?nition: A×B = ABsin? n, where n is a unit vector perpendicular to the
plane of A and B (right-hand rule).
• Properties:
– Result is a vector.
– Magnitude: |A×B| = ABsin?.
– In components: A×B = (A
y
B
x
-A
x
B
y
)k (for 2D vectors in xy-plane).
– A×A = 0; i×j =k, j×k =i, k×i =j.
Resolution of Vectors
• A vector A can be resolved: A
x
= Acos?, A
y
= Asin?, where ? is the angle with
the x-axis.
• In 3D: A = A
x
i+A
y
j+A
z
k.
2
Page 3


Vectors Cheat Sheet
Introduction
Vectors is a fundamental chapter in Physics, essential for understanding quantities with
both magnitude and direction. It forms the basis for analyzing motion, forces, and
other physical phenomena in multiple dimensions, making it crucial for solving complex
problems in various physics exams.
Scalars and Vectors
Type Description Examples
Scalar Quantities with only magnitude,
no direction.
Mass, time, speed, energy
Vector Quantities with both magnitude
and direction.
Displacement, velocity,
force, acceleration
Representation of Vectors
• Notation: Vectors are denoted by bold letters (e.g., A) or with an arrow (e.g.,
- ?
A).
• Components: A vector in 2D: A = A
x
i+A
y
j, where i, j are unit vectors along x
and y axes.
• Magnitude: |A| =
v
A
2
x
+A
2
y
for a 2D vector.
• Direction: ? = tan
-1
(A
y
/A
x
), measured from the positive x-axis.
Types of Vectors
Type Description
Unit Vector Vector with magnitude 1, e.g., ˆ a =A/|A|.
Zero Vector Vector with zero magnitude, no speci?c direction.
Equal Vectors Same magnitude and direction, e.g., A =B.
Parallel Vectors Same or opposite direction, di?erent magnitudes.
1
Vector Addition
• Parallelogram Law: If A and B are adjacent sides of a parallelogram, resultant
R is the diagonal.
– Magnitude: |R| =
v
A
2
+B
2
+2ABcos?, where ? is the angle between A
and B.
– Direction: tan? = (Bsin?)/(A+Bcos?).
• Triangle Law: Place tail of B at head of A; R is from tail of A to head of B.
• Component Method: R =A+B = (A
x
+B
x
)i+(A
y
+B
y
)j.
Vector Subtraction
• A-B =A+(-B), where-B is B reversed in direction.
• Magnitude and direction calculated similarly to addition.
Dot Product (Scalar Product)
• De?nition: A·B = ABcos?, where ? is the angle between A and B.
• Properties:
– Result is a scalar.
– A·B = A
x
B
x
+A
y
B
y
(in component form).
– A·A = A
2
; i·i =j·j = 1; i·j = 0.
Cross Product (Vector Product)
• De?nition: A×B = ABsin? n, where n is a unit vector perpendicular to the
plane of A and B (right-hand rule).
• Properties:
– Result is a vector.
– Magnitude: |A×B| = ABsin?.
– In components: A×B = (A
y
B
x
-A
x
B
y
)k (for 2D vectors in xy-plane).
– A×A = 0; i×j =k, j×k =i, k×i =j.
Resolution of Vectors
• A vector A can be resolved: A
x
= Acos?, A
y
= Asin?, where ? is the angle with
the x-axis.
• In 3D: A = A
x
i+A
y
j+A
z
k.
2
Key Formulas and Concepts
• Position Vector: r = xi+yj+zk.
• Displacement Vector: ?r =r
2
-r
1
.
• Angle Between Vectors: cos? = (A·B)/(AB).
• Projection of A on B: (A·B)/B.
• Area of Parallelogram: |A×B| gives the area formed by A and B.
Solved Examples
1. Resultant Vector: Two vectors A = 3i+4j and B = 5i-2j are given. Find the
magnitude and direction of their resultant.
• Solution:
– R =A+B = (3+5)i+(4-2)j = 8i+2j.
– Magnitude: |R| =
v
8
2
+2
2
=
v
64+4 =
v
68˜ 8.25.
– Direction: ? = tan
-1
(R
y
/R
x
) = tan
-1
(2/8) = tan
-1
(0.25)˜ 14.04
?
with
x-axis.
– Answer: Magnitude˜ 8.25, ?˜ 14.04
?
.
2. Dot Product Application: Find the angle between vectors A = 2i+3j+k and
B =i-j+2k.
• Solution:
– A·B = (2)(1)+(3)(-1)+(1)(2) = 2-3+2 = 1.
– |A| =
v
2
2
+3
2
+1
2
=
v
4+9+1 =
v
14.
– |B| =
v
1
2
+(-1)
2
+2
2
=
v
1+1+4 =
v
6.
– cos? = (A·B)/(|A||B|) = 1/(
v
14·
v
6) = 1/
v
84˜ 0.1091.
– ? = cos
-1
(0.1091)˜ 83.73
?
.
– Answer: ?˜ 83.73
?
.
3. Cross Product Application: Calculate the area of the parallelogram formed by
vectors A = 4i+2j and B = 3i+5j.
• Solution:
– A×B = (A
y
B
x
-A
x
B
y
)k = (2·3-4·5)k = (6-20)k =-14k.
– Magnitude: |A×B| =|-14| = 14.
– Area of parallelogram =|A×B| = 14 square units.
– Answer: Area = 14 square units.
3
Page 4


Vectors Cheat Sheet
Introduction
Vectors is a fundamental chapter in Physics, essential for understanding quantities with
both magnitude and direction. It forms the basis for analyzing motion, forces, and
other physical phenomena in multiple dimensions, making it crucial for solving complex
problems in various physics exams.
Scalars and Vectors
Type Description Examples
Scalar Quantities with only magnitude,
no direction.
Mass, time, speed, energy
Vector Quantities with both magnitude
and direction.
Displacement, velocity,
force, acceleration
Representation of Vectors
• Notation: Vectors are denoted by bold letters (e.g., A) or with an arrow (e.g.,
- ?
A).
• Components: A vector in 2D: A = A
x
i+A
y
j, where i, j are unit vectors along x
and y axes.
• Magnitude: |A| =
v
A
2
x
+A
2
y
for a 2D vector.
• Direction: ? = tan
-1
(A
y
/A
x
), measured from the positive x-axis.
Types of Vectors
Type Description
Unit Vector Vector with magnitude 1, e.g., ˆ a =A/|A|.
Zero Vector Vector with zero magnitude, no speci?c direction.
Equal Vectors Same magnitude and direction, e.g., A =B.
Parallel Vectors Same or opposite direction, di?erent magnitudes.
1
Vector Addition
• Parallelogram Law: If A and B are adjacent sides of a parallelogram, resultant
R is the diagonal.
– Magnitude: |R| =
v
A
2
+B
2
+2ABcos?, where ? is the angle between A
and B.
– Direction: tan? = (Bsin?)/(A+Bcos?).
• Triangle Law: Place tail of B at head of A; R is from tail of A to head of B.
• Component Method: R =A+B = (A
x
+B
x
)i+(A
y
+B
y
)j.
Vector Subtraction
• A-B =A+(-B), where-B is B reversed in direction.
• Magnitude and direction calculated similarly to addition.
Dot Product (Scalar Product)
• De?nition: A·B = ABcos?, where ? is the angle between A and B.
• Properties:
– Result is a scalar.
– A·B = A
x
B
x
+A
y
B
y
(in component form).
– A·A = A
2
; i·i =j·j = 1; i·j = 0.
Cross Product (Vector Product)
• De?nition: A×B = ABsin? n, where n is a unit vector perpendicular to the
plane of A and B (right-hand rule).
• Properties:
– Result is a vector.
– Magnitude: |A×B| = ABsin?.
– In components: A×B = (A
y
B
x
-A
x
B
y
)k (for 2D vectors in xy-plane).
– A×A = 0; i×j =k, j×k =i, k×i =j.
Resolution of Vectors
• A vector A can be resolved: A
x
= Acos?, A
y
= Asin?, where ? is the angle with
the x-axis.
• In 3D: A = A
x
i+A
y
j+A
z
k.
2
Key Formulas and Concepts
• Position Vector: r = xi+yj+zk.
• Displacement Vector: ?r =r
2
-r
1
.
• Angle Between Vectors: cos? = (A·B)/(AB).
• Projection of A on B: (A·B)/B.
• Area of Parallelogram: |A×B| gives the area formed by A and B.
Solved Examples
1. Resultant Vector: Two vectors A = 3i+4j and B = 5i-2j are given. Find the
magnitude and direction of their resultant.
• Solution:
– R =A+B = (3+5)i+(4-2)j = 8i+2j.
– Magnitude: |R| =
v
8
2
+2
2
=
v
64+4 =
v
68˜ 8.25.
– Direction: ? = tan
-1
(R
y
/R
x
) = tan
-1
(2/8) = tan
-1
(0.25)˜ 14.04
?
with
x-axis.
– Answer: Magnitude˜ 8.25, ?˜ 14.04
?
.
2. Dot Product Application: Find the angle between vectors A = 2i+3j+k and
B =i-j+2k.
• Solution:
– A·B = (2)(1)+(3)(-1)+(1)(2) = 2-3+2 = 1.
– |A| =
v
2
2
+3
2
+1
2
=
v
4+9+1 =
v
14.
– |B| =
v
1
2
+(-1)
2
+2
2
=
v
1+1+4 =
v
6.
– cos? = (A·B)/(|A||B|) = 1/(
v
14·
v
6) = 1/
v
84˜ 0.1091.
– ? = cos
-1
(0.1091)˜ 83.73
?
.
– Answer: ?˜ 83.73
?
.
3. Cross Product Application: Calculate the area of the parallelogram formed by
vectors A = 4i+2j and B = 3i+5j.
• Solution:
– A×B = (A
y
B
x
-A
x
B
y
)k = (2·3-4·5)k = (6-20)k =-14k.
– Magnitude: |A×B| =|-14| = 14.
– Area of parallelogram =|A×B| = 14 square units.
– Answer: Area = 14 square units.
3
4. Vector Resolution: A vector A has a magnitude of 10 units and makes an angle
of 30
?
with the positive x-axis. Find its components.
• Solution:
– A
x
= Acos? = 10cos30
?
= 10·(
v
3/2) = 5
v
3˜ 8.66.
– A
y
= Asin? = 10sin30
?
= 10·(1/2) = 5.
– Answer: A = 8.66i+5j.
5. Projection of Vector: Find the projection ofA = 2i+3j-k onB =i+2j+2k.
• Solution:
– Projection = (A·B)/|B|.
– A·B = (2)(1)+(3)(2)+(-1)(2) = 2+6-2 = 6.
– |B| =
v
1
2
+2
2
+2
2
=
v
1+4+4 =
v
9 = 3.
– Projection = 6/3 = 2.
– Answer: Projection = 2 units.
4