Solved Examples Vectors - AP Precalculus - Grade 10 PDF Download

 Page 1


Basics of Vectors
Example 1:
Question: Dene a vector and list three dierent types of vectors. Provide an
example for each type.
Solution: A vector is a mathematical quantity that has both magnitude
and direction.
Types of vectors: 1. Position Vector: Represents the position of a point
in space relative to an origin. Example: ~ r = 2
^
i 3
^
j + 5
^
k
2. DisplacementVector: Represents the change in position from an initial
point to a nal point. Example: If an object moves from point A to point B,
the displacement vector
~
d is
~
AB.
3. Free Vector: Has a xed magnitude and direction in space. Example:
~ v = 4
^
i + 2
^
j
Example 2:
Question: What are the properties of vectors? Explain three fundamental
properties.
Solution: Vectors have several properties. Here are three fundamental
properties:
1. Closure under Addition and Scalar Multiplication: Vectors are
closed under addition and scalar multiplication, meaning that the sum of two
vectors and the product of a vector by a scalar is also a vector.
2. Associativity of Vector Addition: Vector addition is associative,
which means that the grouping of vectors in an addition operation does not
aect the result.
3. CommutativityofVectorAddition: Vector addition is commutative,
meaning that the order in which vectors are added does not aect the result.
Example 3:
Question: Explain the concept of a unit vector. Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. It is often used
to represent direction without considering scale.
Example: The unit vector in the direction of the positive x-axis is
^
i = 1
^
i.
Example 4:
Question: What is the signicance of the direction in vectors? Provide an
example.
Solution: Direction in vectors indicates the orientation or path along which
the vector points. It provides crucial information about how the vector is
oriented in space.
Example: Consider a velocity vector ~ v = 20
^
i + 10
^
j representing a car's
velocity. The direction, given by the unit vector, indicates the car's heading.
1
Page 2


Basics of Vectors
Example 1:
Question: Dene a vector and list three dierent types of vectors. Provide an
example for each type.
Solution: A vector is a mathematical quantity that has both magnitude
and direction.
Types of vectors: 1. Position Vector: Represents the position of a point
in space relative to an origin. Example: ~ r = 2
^
i 3
^
j + 5
^
k
2. DisplacementVector: Represents the change in position from an initial
point to a nal point. Example: If an object moves from point A to point B,
the displacement vector
~
d is
~
AB.
3. Free Vector: Has a xed magnitude and direction in space. Example:
~ v = 4
^
i + 2
^
j
Example 2:
Question: What are the properties of vectors? Explain three fundamental
properties.
Solution: Vectors have several properties. Here are three fundamental
properties:
1. Closure under Addition and Scalar Multiplication: Vectors are
closed under addition and scalar multiplication, meaning that the sum of two
vectors and the product of a vector by a scalar is also a vector.
2. Associativity of Vector Addition: Vector addition is associative,
which means that the grouping of vectors in an addition operation does not
aect the result.
3. CommutativityofVectorAddition: Vector addition is commutative,
meaning that the order in which vectors are added does not aect the result.
Example 3:
Question: Explain the concept of a unit vector. Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. It is often used
to represent direction without considering scale.
Example: The unit vector in the direction of the positive x-axis is
^
i = 1
^
i.
Example 4:
Question: What is the signicance of the direction in vectors? Provide an
example.
Solution: Direction in vectors indicates the orientation or path along which
the vector points. It provides crucial information about how the vector is
oriented in space.
Example: Consider a velocity vector ~ v = 20
^
i + 10
^
j representing a car's
velocity. The direction, given by the unit vector, indicates the car's heading.
1
Example 5:
Question: Dene a scalar and give an example in the context of vectors.
Solution: A scalar is a quantity that only has magnitude and no direction.
Example: In the vector
~
F = 50
^
i 30
^
j, the scalar values are 50 and -30,
representing the magnitudes of the vector components in the x and y directions,
respectively.
Addition of Vectors
Example 1:
Question: Given vectors
~
P = 2i + 3jk and
~
Q =i + 4j + 2k, nd the vector
~
R such that 2
~
P +
~
R = 3
~
Q.
Solution:
2
~
P +
~
R = 3
~
Q;
2(2i + 3jk) +
~
R = 3(i + 4j + 2k);
4i + 6j 2k +
~
R =3i + 12j + 6k;
~
R =7i + 6j + 8k:
Example 2:
Question: Three forces
~
F
1
,
~
F
2
, and
~
F
3
act on a point. If
~
F
1
= 3i 4j + 2k,
~
F
2
=2i + 5j 3k, and
~
F
3
=i 2j + 4k, nd the resultant force vector.
Solution:
~
R =
~
F
1
+
~
F
2
+
~
F
3
;
~
R = (3i 4j + 2k) + (2i + 5j 3k) + (i 2j + 4k);
~
R =i +ij + 3k;
~
R = 2ij + 3k:
Example 3:
Question: Dene a displacement vector and give an example.
Solution: A displacement vector represents the change in position from an
initial point to a nal point. For example, if an object moves from point A to
point B, the displacement vector
~
d is
~
AB.
Example 4:
Question: What is a unit vector? Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. For example,
the unit vector in the direction of the positive x-axis is
^
i = 1i.
2
Page 3


Basics of Vectors
Example 1:
Question: Dene a vector and list three dierent types of vectors. Provide an
example for each type.
Solution: A vector is a mathematical quantity that has both magnitude
and direction.
Types of vectors: 1. Position Vector: Represents the position of a point
in space relative to an origin. Example: ~ r = 2
^
i 3
^
j + 5
^
k
2. DisplacementVector: Represents the change in position from an initial
point to a nal point. Example: If an object moves from point A to point B,
the displacement vector
~
d is
~
AB.
3. Free Vector: Has a xed magnitude and direction in space. Example:
~ v = 4
^
i + 2
^
j
Example 2:
Question: What are the properties of vectors? Explain three fundamental
properties.
Solution: Vectors have several properties. Here are three fundamental
properties:
1. Closure under Addition and Scalar Multiplication: Vectors are
closed under addition and scalar multiplication, meaning that the sum of two
vectors and the product of a vector by a scalar is also a vector.
2. Associativity of Vector Addition: Vector addition is associative,
which means that the grouping of vectors in an addition operation does not
aect the result.
3. CommutativityofVectorAddition: Vector addition is commutative,
meaning that the order in which vectors are added does not aect the result.
Example 3:
Question: Explain the concept of a unit vector. Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. It is often used
to represent direction without considering scale.
Example: The unit vector in the direction of the positive x-axis is
^
i = 1
^
i.
Example 4:
Question: What is the signicance of the direction in vectors? Provide an
example.
Solution: Direction in vectors indicates the orientation or path along which
the vector points. It provides crucial information about how the vector is
oriented in space.
Example: Consider a velocity vector ~ v = 20
^
i + 10
^
j representing a car's
velocity. The direction, given by the unit vector, indicates the car's heading.
1
Example 5:
Question: Dene a scalar and give an example in the context of vectors.
Solution: A scalar is a quantity that only has magnitude and no direction.
Example: In the vector
~
F = 50
^
i 30
^
j, the scalar values are 50 and -30,
representing the magnitudes of the vector components in the x and y directions,
respectively.
Addition of Vectors
Example 1:
Question: Given vectors
~
P = 2i + 3jk and
~
Q =i + 4j + 2k, nd the vector
~
R such that 2
~
P +
~
R = 3
~
Q.
Solution:
2
~
P +
~
R = 3
~
Q;
2(2i + 3jk) +
~
R = 3(i + 4j + 2k);
4i + 6j 2k +
~
R =3i + 12j + 6k;
~
R =7i + 6j + 8k:
Example 2:
Question: Three forces
~
F
1
,
~
F
2
, and
~
F
3
act on a point. If
~
F
1
= 3i 4j + 2k,
~
F
2
=2i + 5j 3k, and
~
F
3
=i 2j + 4k, nd the resultant force vector.
Solution:
~
R =
~
F
1
+
~
F
2
+
~
F
3
;
~
R = (3i 4j + 2k) + (2i + 5j 3k) + (i 2j + 4k);
~
R =i +ij + 3k;
~
R = 2ij + 3k:
Example 3:
Question: Dene a displacement vector and give an example.
Solution: A displacement vector represents the change in position from an
initial point to a nal point. For example, if an object moves from point A to
point B, the displacement vector
~
d is
~
AB.
Example 4:
Question: What is a unit vector? Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. For example,
the unit vector in the direction of the positive x-axis is
^
i = 1i.
2
Product of Vectors
Example 1:
Question: Given vectors
~
X =i + 2j 3k and
~
Y = 2ij + 4k, nd the angle
between the vectors using the vector product.
Solution:
~
X
~
Y =






^
i
^
j
^
k
1 2 3
2 1 4






;
~
X
~
Y =
^
i(2
 4 (3)
 (1))
^
j(1
 4 (3)
 2) +
^
k(1
 (1) 2
 2);
~
X
~
Y = 5
^
i 10
^
j 5
^
k;
j
~
X
~
Yj =
p
5
2
+ (10)
2
+ (5)
2
;
j
~
X
~
Yj =
p
25 + 100 + 25;
j
~
X
~
Yj =
p
150;
j
~
Xj =
p
1
2
+ 2
2
+ (3)
2
;
j
~
Xj =
p
1 + 4 + 9;
j
~
Xj =
p
14;
j
~
Yj =
p
2
2
+ (1)
2
+ 4
2
;
j
~
Yj =
p
4 + 1 + 16;
j
~
Yj =
p
21;
sin() =
j
~
X
~
Yj
j
~
Xjj
~
Yj
=
p
150
p
14

p
21
;
 = sin
1
 
p
150
p
14

p
21
!
:
Example 2:
Question: Find the area of the parallelogram formed by vectors
~
U = 3i+2jk
and
~
V =i + 4j + 2k.
3
Page 4


Basics of Vectors
Example 1:
Question: Dene a vector and list three dierent types of vectors. Provide an
example for each type.
Solution: A vector is a mathematical quantity that has both magnitude
and direction.
Types of vectors: 1. Position Vector: Represents the position of a point
in space relative to an origin. Example: ~ r = 2
^
i 3
^
j + 5
^
k
2. DisplacementVector: Represents the change in position from an initial
point to a nal point. Example: If an object moves from point A to point B,
the displacement vector
~
d is
~
AB.
3. Free Vector: Has a xed magnitude and direction in space. Example:
~ v = 4
^
i + 2
^
j
Example 2:
Question: What are the properties of vectors? Explain three fundamental
properties.
Solution: Vectors have several properties. Here are three fundamental
properties:
1. Closure under Addition and Scalar Multiplication: Vectors are
closed under addition and scalar multiplication, meaning that the sum of two
vectors and the product of a vector by a scalar is also a vector.
2. Associativity of Vector Addition: Vector addition is associative,
which means that the grouping of vectors in an addition operation does not
aect the result.
3. CommutativityofVectorAddition: Vector addition is commutative,
meaning that the order in which vectors are added does not aect the result.
Example 3:
Question: Explain the concept of a unit vector. Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. It is often used
to represent direction without considering scale.
Example: The unit vector in the direction of the positive x-axis is
^
i = 1
^
i.
Example 4:
Question: What is the signicance of the direction in vectors? Provide an
example.
Solution: Direction in vectors indicates the orientation or path along which
the vector points. It provides crucial information about how the vector is
oriented in space.
Example: Consider a velocity vector ~ v = 20
^
i + 10
^
j representing a car's
velocity. The direction, given by the unit vector, indicates the car's heading.
1
Example 5:
Question: Dene a scalar and give an example in the context of vectors.
Solution: A scalar is a quantity that only has magnitude and no direction.
Example: In the vector
~
F = 50
^
i 30
^
j, the scalar values are 50 and -30,
representing the magnitudes of the vector components in the x and y directions,
respectively.
Addition of Vectors
Example 1:
Question: Given vectors
~
P = 2i + 3jk and
~
Q =i + 4j + 2k, nd the vector
~
R such that 2
~
P +
~
R = 3
~
Q.
Solution:
2
~
P +
~
R = 3
~
Q;
2(2i + 3jk) +
~
R = 3(i + 4j + 2k);
4i + 6j 2k +
~
R =3i + 12j + 6k;
~
R =7i + 6j + 8k:
Example 2:
Question: Three forces
~
F
1
,
~
F
2
, and
~
F
3
act on a point. If
~
F
1
= 3i 4j + 2k,
~
F
2
=2i + 5j 3k, and
~
F
3
=i 2j + 4k, nd the resultant force vector.
Solution:
~
R =
~
F
1
+
~
F
2
+
~
F
3
;
~
R = (3i 4j + 2k) + (2i + 5j 3k) + (i 2j + 4k);
~
R =i +ij + 3k;
~
R = 2ij + 3k:
Example 3:
Question: Dene a displacement vector and give an example.
Solution: A displacement vector represents the change in position from an
initial point to a nal point. For example, if an object moves from point A to
point B, the displacement vector
~
d is
~
AB.
Example 4:
Question: What is a unit vector? Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. For example,
the unit vector in the direction of the positive x-axis is
^
i = 1i.
2
Product of Vectors
Example 1:
Question: Given vectors
~
X =i + 2j 3k and
~
Y = 2ij + 4k, nd the angle
between the vectors using the vector product.
Solution:
~
X
~
Y =






^
i
^
j
^
k
1 2 3
2 1 4






;
~
X
~
Y =
^
i(2
 4 (3)
 (1))
^
j(1
 4 (3)
 2) +
^
k(1
 (1) 2
 2);
~
X
~
Y = 5
^
i 10
^
j 5
^
k;
j
~
X
~
Yj =
p
5
2
+ (10)
2
+ (5)
2
;
j
~
X
~
Yj =
p
25 + 100 + 25;
j
~
X
~
Yj =
p
150;
j
~
Xj =
p
1
2
+ 2
2
+ (3)
2
;
j
~
Xj =
p
1 + 4 + 9;
j
~
Xj =
p
14;
j
~
Yj =
p
2
2
+ (1)
2
+ 4
2
;
j
~
Yj =
p
4 + 1 + 16;
j
~
Yj =
p
21;
sin() =
j
~
X
~
Yj
j
~
Xjj
~
Yj
=
p
150
p
14

p
21
;
 = sin
1
 
p
150
p
14

p
21
!
:
Example 2:
Question: Find the area of the parallelogram formed by vectors
~
U = 3i+2jk
and
~
V =i + 4j + 2k.
3
Solution:
~
U
~
V =






^
i
^
j
^
k
3 2 1
1 4 2






;
~
U
~
V =
^
i((2)(2) (1)(4))
^
j((3)(2) (1)(1)) +
^
k((3)(4) (2)(1));
~
U
~
V = 8
^
i 5
^
j + 14
^
k;
j
~
U
~
Vj =
p
8
2
+ (5)
2
+ 14
2
;
j
~
U
~
Vj =
p
64 + 25 + 196;
j
~
U
~
Vj =
p
285:
So, the area of the parallelogram isj
~
U
~
Vj =
p
285.
Example 3:
Question: Given vectors
~
M = i 3j + 2k and
~
N = 2i +j 4k, determine
whether the vectors are parallel, perpendicular, or neither.
Solution:
~
M

~
N = (1)(2) + (3)(1) + (2)(4) = 2 3 8 =9;
j
~
Mj =
p
1
2
+ (3)
2
+ 2
2
=
p
1 + 9 + 4 =
p
14;
j
~
Nj =
p
2
2
+ 1
2
+ (4)
2
=
p
4 + 1 + 16 =
p
21:
Since
~
M

~
N6= 0 andj
~
Mj6= 0 andj
~
Nj6= 0, the vectors are neither parallel
nor perpendicular.
Resolution of Vectors
Example 1:
Question: A force
~
F is applied at an angle of 30

to the horizontal. Resolve
the force into horizontal and vertical components.
Solution:
Horizontal component:
~
F
x
=
~
F
cos(30

); Vertical component:
~
F
y
=
~
F
sin(30

):
Example 2:
Question: For vectors
~
P = 3i + 2j and
~
Q =i + 4j, nd the projection of
~
P
onto
~
Q.
Solution:
Scalar projection: proj
~
Q
(
~
P ) =
~
P

~
Q
j
~
Qj
; Vector projection: proj
~
Q
(
~
P ) =
~
P

~
Q
j
~
Qj
2


~
Q:
4
Page 5


Basics of Vectors
Example 1:
Question: Dene a vector and list three dierent types of vectors. Provide an
example for each type.
Solution: A vector is a mathematical quantity that has both magnitude
and direction.
Types of vectors: 1. Position Vector: Represents the position of a point
in space relative to an origin. Example: ~ r = 2
^
i 3
^
j + 5
^
k
2. DisplacementVector: Represents the change in position from an initial
point to a nal point. Example: If an object moves from point A to point B,
the displacement vector
~
d is
~
AB.
3. Free Vector: Has a xed magnitude and direction in space. Example:
~ v = 4
^
i + 2
^
j
Example 2:
Question: What are the properties of vectors? Explain three fundamental
properties.
Solution: Vectors have several properties. Here are three fundamental
properties:
1. Closure under Addition and Scalar Multiplication: Vectors are
closed under addition and scalar multiplication, meaning that the sum of two
vectors and the product of a vector by a scalar is also a vector.
2. Associativity of Vector Addition: Vector addition is associative,
which means that the grouping of vectors in an addition operation does not
aect the result.
3. CommutativityofVectorAddition: Vector addition is commutative,
meaning that the order in which vectors are added does not aect the result.
Example 3:
Question: Explain the concept of a unit vector. Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. It is often used
to represent direction without considering scale.
Example: The unit vector in the direction of the positive x-axis is
^
i = 1
^
i.
Example 4:
Question: What is the signicance of the direction in vectors? Provide an
example.
Solution: Direction in vectors indicates the orientation or path along which
the vector points. It provides crucial information about how the vector is
oriented in space.
Example: Consider a velocity vector ~ v = 20
^
i + 10
^
j representing a car's
velocity. The direction, given by the unit vector, indicates the car's heading.
1
Example 5:
Question: Dene a scalar and give an example in the context of vectors.
Solution: A scalar is a quantity that only has magnitude and no direction.
Example: In the vector
~
F = 50
^
i 30
^
j, the scalar values are 50 and -30,
representing the magnitudes of the vector components in the x and y directions,
respectively.
Addition of Vectors
Example 1:
Question: Given vectors
~
P = 2i + 3jk and
~
Q =i + 4j + 2k, nd the vector
~
R such that 2
~
P +
~
R = 3
~
Q.
Solution:
2
~
P +
~
R = 3
~
Q;
2(2i + 3jk) +
~
R = 3(i + 4j + 2k);
4i + 6j 2k +
~
R =3i + 12j + 6k;
~
R =7i + 6j + 8k:
Example 2:
Question: Three forces
~
F
1
,
~
F
2
, and
~
F
3
act on a point. If
~
F
1
= 3i 4j + 2k,
~
F
2
=2i + 5j 3k, and
~
F
3
=i 2j + 4k, nd the resultant force vector.
Solution:
~
R =
~
F
1
+
~
F
2
+
~
F
3
;
~
R = (3i 4j + 2k) + (2i + 5j 3k) + (i 2j + 4k);
~
R =i +ij + 3k;
~
R = 2ij + 3k:
Example 3:
Question: Dene a displacement vector and give an example.
Solution: A displacement vector represents the change in position from an
initial point to a nal point. For example, if an object moves from point A to
point B, the displacement vector
~
d is
~
AB.
Example 4:
Question: What is a unit vector? Provide an example.
Solution: A unit vector is a vector with a magnitude of 1. For example,
the unit vector in the direction of the positive x-axis is
^
i = 1i.
2
Product of Vectors
Example 1:
Question: Given vectors
~
X =i + 2j 3k and
~
Y = 2ij + 4k, nd the angle
between the vectors using the vector product.
Solution:
~
X
~
Y =






^
i
^
j
^
k
1 2 3
2 1 4






;
~
X
~
Y =
^
i(2
 4 (3)
 (1))
^
j(1
 4 (3)
 2) +
^
k(1
 (1) 2
 2);
~
X
~
Y = 5
^
i 10
^
j 5
^
k;
j
~
X
~
Yj =
p
5
2
+ (10)
2
+ (5)
2
;
j
~
X
~
Yj =
p
25 + 100 + 25;
j
~
X
~
Yj =
p
150;
j
~
Xj =
p
1
2
+ 2
2
+ (3)
2
;
j
~
Xj =
p
1 + 4 + 9;
j
~
Xj =
p
14;
j
~
Yj =
p
2
2
+ (1)
2
+ 4
2
;
j
~
Yj =
p
4 + 1 + 16;
j
~
Yj =
p
21;
sin() =
j
~
X
~
Yj
j
~
Xjj
~
Yj
=
p
150
p
14

p
21
;
 = sin
1
 
p
150
p
14

p
21
!
:
Example 2:
Question: Find the area of the parallelogram formed by vectors
~
U = 3i+2jk
and
~
V =i + 4j + 2k.
3
Solution:
~
U
~
V =






^
i
^
j
^
k
3 2 1
1 4 2






;
~
U
~
V =
^
i((2)(2) (1)(4))
^
j((3)(2) (1)(1)) +
^
k((3)(4) (2)(1));
~
U
~
V = 8
^
i 5
^
j + 14
^
k;
j
~
U
~
Vj =
p
8
2
+ (5)
2
+ 14
2
;
j
~
U
~
Vj =
p
64 + 25 + 196;
j
~
U
~
Vj =
p
285:
So, the area of the parallelogram isj
~
U
~
Vj =
p
285.
Example 3:
Question: Given vectors
~
M = i 3j + 2k and
~
N = 2i +j 4k, determine
whether the vectors are parallel, perpendicular, or neither.
Solution:
~
M

~
N = (1)(2) + (3)(1) + (2)(4) = 2 3 8 =9;
j
~
Mj =
p
1
2
+ (3)
2
+ 2
2
=
p
1 + 9 + 4 =
p
14;
j
~
Nj =
p
2
2
+ 1
2
+ (4)
2
=
p
4 + 1 + 16 =
p
21:
Since
~
M

~
N6= 0 andj
~
Mj6= 0 andj
~
Nj6= 0, the vectors are neither parallel
nor perpendicular.
Resolution of Vectors
Example 1:
Question: A force
~
F is applied at an angle of 30

to the horizontal. Resolve
the force into horizontal and vertical components.
Solution:
Horizontal component:
~
F
x
=
~
F
cos(30

); Vertical component:
~
F
y
=
~
F
sin(30

):
Example 2:
Question: For vectors
~
P = 3i + 2j and
~
Q =i + 4j, nd the projection of
~
P
onto
~
Q.
Solution:
Scalar projection: proj
~
Q
(
~
P ) =
~
P

~
Q
j
~
Qj
; Vector projection: proj
~
Q
(
~
P ) =
~
P

~
Q
j
~
Qj
2


~
Q:
4
Scalar projection: proj
~
Q
(
~
P ) =
(3
1) + (2
 4)
p
(1)
2
+ 4
2
;
Scalar projection: proj
~
Q
(
~
P ) =
3 + 8
p
17
;
Scalar projection: proj
~
Q
(
~
P ) =
5
p
17
;
Vector projection: proj
~
Q
(
~
P ) =
5
17
(i + 4j):
Example 3:
Question: Given a vector
~
V = 4i 3j, nd two unit vectors in the direction
of
~
V .
Solution:
Unit vector: ^ u =
~
V
j
~
Vj
:
So, two unit vectors in the direction of
~
V are ^ u
1
and ^ u
2
, where:
^ u
1
=
4i 3j
p
4
2
+ (3)
2
=
4i 3j
5
;
^ u
2
=
4i 3j
5
:
5