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10. VECTORS AND THE GEOMETRY OF SPACE. VECTORS AND THE GEOMETRY OF SPACE. 10.2 Vectors. In this section, we will learn about: Vectors and their applications. VECTOR.

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10

VECTORS AND

THE GEOMETRY OF SPACE


VECTORS AND THE GEOMETRY OF SPACE

10.2Vectors

In this section, we will learn about:

Vectors and their applications.


VECTOR

  • The term vectoris used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction.


REPRESENTING A VECTOR

  • A vector is often represented by an arrow or a directed line segment.

    • The length of the arrow represents the magnitude of the vector.

    • The arrow points in the direction of the vector.


DENOTING A VECTOR

  • We denote a vector by either:

    • Printing a letter in boldface (v)

    • Putting an arrow above the letter ( )


VECTORS

  • For instance, suppose a particle moves along a line segment from point A to point B.


VECTORS

  • The corresponding displacement vector vhas initial pointA(the tail) and terminal pointB(the tip).

    • We indicate this by writing v = .


VECTORS

  • Notice that the vector u = has the same length and the same direction as v even though it is in a different position.

    • We say u and v are equivalent(or equal) and write u = v.


ZERO VECTOR

  • The zero vector, denoted by 0, has length 0.

    • It is the only vector with no specific direction.


COMBINING VECTORS

  • Suppose a particle moves from A to B.

  • So,its displacement vector is .


COMBINING VECTORS

  • Then, the particle changes direction, and moves from B to C—with displacement vector .


COMBINING VECTORS

  • The combined effect of these displacements is that the particle has moved from A to C.


COMBINING VECTORS

  • The resulting displacement vector is called the sumof and .

  • We write:


ADDING VECTORS

  • In general, if we start with vectors u and v, we first move v so that its tail coincides with the tip of u and define the sum of u and vas follows.


VECTOR ADDITION—DEFINITION

  • If u and v are vectors positioned so the initial point of v is at the terminal point of u, then the sum u + v is the vector from the initial point of u to the terminal point of v.


VECTOR ADDITION

  • The definition of vector addition is illustrated here.


TRIANGLE LAW

  • You can see why this definition is sometimes called the Triangle Law.


VECTOR ADDITION

  • Here, we start with the same vectors u and v as earlier and draw another copy of v with the same initial point as u.


VECTOR ADDITION

  • Completing the parallelogram, we see that: u + v = v + u


VECTOR ADDITION

  • This also gives another way to construct the sum:

    • If we place u and v so they start at the same point, then u + v lies along the diagonal of the parallelogram with u and v as sides.


PARALLELOGRAM LAW

  • This is called the Parallelogram Law.


VECTOR ADDITION

Example 1

  • Draw the sum of the vectors a and b shown here.


VECTOR ADDITION

Example 1

  • First, we translate b and place its tail at the tip of a—being careful to draw a copy of b that has the same length and direction.


VECTOR ADDITION

Example 1

  • Then, we draw the vector a + b starting at the initial point of a and ending at the terminal point of the copy of b.


VECTOR ADDITION

Example 1

  • Alternatively, we could place b so it starts where a starts and construct a + b by the Parallelogram Law.


MULTIPLYING VECTORS

  • It is possible to multiply a vector by a real number c.


SCALAR

  • In this context, we call the real number c a scalar—to distinguish it from a vector.


MULTIPLYING SCALARS

  • For instance, we want 2v to be the same vector as v + v, which has the same direction as v but is twice as long.

  • In general, we multiply a vector by a scalar as follows.


SCALAR MULTIPLICATION—DEFINITION

  • If c is a scalar and v is a vector, the scalar multiplecv is:

  • The vector whose length is |c| times the length of v and whose direction is the same as v if c > 0 and is opposite to v if c < 0.

    • If c = 0 or v = 0, then cv = 0.


SCALAR MULTIPLICATION

  • The definition is illustrated here.

    • We see that real numbers work like scaling factors here.

    • That’s why we call them scalars.


SCALAR MULTIPLICATION

  • Notice that two nonzero vectors are parallelif they are scalar multiples of one another.


SCALAR MULTIPLICATION

  • In particular, the vector –v = (–1)v has the same length as v but points in the opposite direction.

    • We call it the negativeof v.


SUBTRACTING VECTORS

  • By the difference u – v of two vectors, we mean: u – v = u + (–v)


SUBTRACTING VECTORS

  • So, we can construct u – v by first drawing the negative of v, –v, and then adding it to u by the Parallelogram Law.


SUBTRACTING VECTORS

  • Alternatively, since v + (u – v) = u, the vector u – v, when added to v, gives u.


SUBTRACTING VECTORS

  • So, we could construct u by means of the Triangle Law.


SUBTRACTING VECTORS

Example 2

  • If a and b are the vectors shown here, draw a – 2b.


SUBTRACTING VECTORS

Example 2

  • First, we draw the vector –2b pointing in the direction opposite to b and twice as long.

  • Next, we place it with its tail at the tip of a.


SUBTRACTING VECTORS

Example 2

  • Finally, we use the Triangle Law to draw a + (–2b).


COMPONENTS

  • For some purposes, it’s best to introduce a coordinate system and treat vectors algebraically.


COMPONENTS

  • Let’s place the initial point of a vector aat the origin of a rectangular coordinate system.


COMPONENTS

  • Then, the terminal point of a has coordinates of the form (a1, a2) or(a1, a2, a3).

    • This depends on whether our coordinate system is two- or three-dimensional.


COMPONENTS

  • These coordinates are called the componentsof a and we write: a = ‹a1, a2› or a = ‹a1, a2, a3›


COMPONENTS

  • We use the notation ‹a1, a2› for the ordered pair that refers to a vector so as not to confuse it with the ordered pair (a1, a2) that refers to a point in the plane.


COMPONENTS

  • For instance, the vectors shown here are all equivalent to the vector whose terminal point is P(3, 2).


COMPONENTS

  • What they have in common is that the terminal point is reached from the initial point by a displacement of three units to the right and two upward.


COMPONENTS

  • We can think of all these geometric vectors as representationsof the algebraic vector a = ‹3, 2›.


Position vector
POSITION VECTOR

  • The particular representation from the origin to the point P(3, 2) is called the position vectorof the point P.


Position vector1
POSITION VECTOR

  • In three dimensions, the vector a = = ‹a1, a2, a3›is the position vectorof the point P(a1, a2, a3).


Components
COMPONENTS

  • Let’s consider any other representation of a, where the initial point is A(x1, y1, z1) and the terminal point is B(x2, y2, z2).


Components1
COMPONENTS

  • Then, we must have: x1 + a1 = x2, y1 + a2 = y2, z1 + a3 = z2

  • Thus, a1 = x2 – x1, a2 = y2 – y1, a3 = z2 – z1

    • Thus, we have the following result.


Components2
COMPONENTS

Equation 1

  • Given the points A(x1, y1, z1) and B(x2, y2, z2), the vector a with representation is: a = ‹x2 – x1, y2 – y1, z2 – z1›


Components3
COMPONENTS

Example 3

  • Find the vector represented by the directed line segment with initial point A(2, –3, 4) and terminal point B(–2, 1, 1).

    • By Equation 1, the vector corresponding to is: a = ‹–2 –2, 1 – (–3), 1 – 4› = ‹–4, 4, –3›


Length of vector
LENGTH OF VECTOR

  • The magnitude or length of the vector vis the length of any of its representations.

    • It is denoted by the symbol |v| or║v║.


Length of vector1
LENGTH OF VECTOR

  • By using the distance formula to compute the length of a segment , we obtain the following formulas.


Length of 2 d vector
LENGTH OF 2-D VECTOR

  • The length of the two-dimensional (2-D) vector a = ‹a1, a2› is:


Length of 3 d vector
LENGTH OF 3-D VECTOR

  • The length of the three-dimensional (3-D) vector a = ‹a1, a2, a3› is:


Algebraic vectors
ALGEBRAIC VECTORS

  • How do we add vectors algebraically?


Adding algebraic vectors
ADDING ALGEBRAIC VECTORS

  • The figure shows that, if a = ‹a1, a2›and b = ‹b1, b2›, then the sum is a + b = ‹a1 + b1, a2 + b2›at least for the case where the components are positive.


Adding algebraic vectors1
ADDING ALGEBRAIC VECTORS

  • In other words, to add algebraic vectors, we add their components.


Subtracting algebraic vectors
SUBTRACTING ALGEBRAIC VECTORS

  • Similarly, to subtract vectors, we subtract components.


Algebraic vectors1
ALGEBRAIC VECTORS

  • From the similar triangles in the figure, we see that the components of ca are ca1 and ca2.


Multiplying algebraic vectors
MULTIPLYING ALGEBRAIC VECTORS

  • So, to multiply a vector by a scalar, we multiply each component by that scalar.


2 d algebraic vectors
2-D ALGEBRAIC VECTORS

  • If a = ‹a1, a2›and b = ‹b1, b2›, then

  • a + b = ‹a1 + b1, a2 +b2›

  • a – b = ‹a1–b1 , a2–b2›

  • ca = ‹ca1, ca2›


3 d algebraic vectors
3-D ALGEBRAIC VECTORS

  • Similarly, for 3-D vectors,


Algebraic vectors2
ALGEBRAIC VECTORS

Example 4

  • If a = ‹4, 0, 3›and b = ‹–2, 1, 5›, find: |a| and the vectors a + b, a – b, 3b, 2a + 5b


Algebraic vectors3
ALGEBRAIC VECTORS

Example 4

|a|






Components4
COMPONENTS

  • We denote:

    • V2 as the set of all 2-D vectors

    • V3 as the set of all 3-D vectors


Components5
COMPONENTS

  • More generally, we will later need to consider the set Vn of all n-dimensional vectors.

    • An n-dimensional vector is an ordered n-tuplea = ‹a1, a2, …, an›where a1, a2, …, an are real numbers that are called the components of a.


Components6
COMPONENTS

  • Addition and scalar multiplication are defined in terms of components just as for the cases n = 2 and n = 3.


Properties of vectors
PROPERTIES OF VECTORS

  • If a, b, and c are vectors in Vn and c and dare scalars, then


Properties of vectors1
PROPERTIES OF VECTORS

  • These eight properties of vectors can be readily verified either geometrically or algebraically.


Property 1
PROPERTY 1

  • For instance, Property 1 can be seen from this earlier figure.

    • It’s equivalent to the Parallelogram Law.


Property 11
PROPERTY 1

  • It can also be seen as follows for the case n = 2:

  • a + b = ‹a1, a2› + ‹b1, b2› = ‹a1 + b1, a2 + b2›

  • = ‹b1 + a1, b2 + a2› = ‹b1, b2› + ‹a1, a2›

  • = b + a


Property 2
PROPERTY 2

  • We can see why Property 2 (the associative law) is true by looking at this figure and applying the Triangle Law several times, as follows.


Property 21
PROPERTY 2

  • The vector is obtained either by first constructing a + b and then adding c or by adding a to the vector b + c.


Vectors in v 3
VECTORS IN V3

  • Three vectors in V3 play a special role.


Vectors in v 31
VECTORS IN V3

  • Let

  • i = ‹1, 0, 0›

  • j = ‹0, 1, 0›

  • k = ‹0, 0, 1›


Standard basis vectors
STANDARD BASIS VECTORS

  • These vectors i, j, and k are called the standard basis vectors.

    • They have length 1 and point in the directions of the positive x-, y-, and z-axes.


Standard basis vectors1
STANDARD BASIS VECTORS

  • Similarly, in two dimensions, we define:

  • i = ‹1, 0›

  • j = ‹0, 1›


Standard basis vectors2
STANDARD BASIS VECTORS

  • If a = ‹a1, a2, a3›, then we can write:



Standard basis vectors4
STANDARD BASIS VECTORS

  • Thus, any vector in V3 can be expressed in terms of i, j, and k.


Standard basis vectors5
STANDARD BASIS VECTORS

  • For instance, ‹1, –2, 6› = i– 2j + 6k


Standard basis vectors6
STANDARD BASIS VECTORS

Equation 3

  • Similarly, in two dimensions, we can write: a = ‹a1, a2› = a1i + a2j


Components7
COMPONENTS

  • Compare the geometric interpretation of Equations 2 and 3 with the earlier figures.


Components8
COMPONENTS

Example 5

  • If a = i + 2j – 3k and b = 4i + 7k, express the vector 2a + 3b in terms of i, j, and k.


Components9
COMPONENTS

Example 5

  • Using Properties 1, 2, 5, 6, and 7 of vectors, we have: 2a + 3b = 2(i + 2j – 3k) + 3(4i + 7k) = 2i + 4j – 6k + 12i + 21k = 14i + 4j + 15k


Unit vector
UNIT VECTOR

  • A unit vector is a vector whose length is 1.

    • For instance, i, j, and k are all unit vectors.


Unit vectors
UNIT VECTORS

Equation 4

  • In general, if a≠ 0, then the unit vector that has the same direction as a is:


Unit vectors1
UNIT VECTORS

  • In order to verify this, we let c = 1/|a|.

    • Then, u = ca and c is a positive scalar; so, u has the same direction as a.

    • Also,


Unit vectors2
UNIT VECTORS

Example 6

  • Find the unit vector in the direction of the vector 2i – j – 2k.


Unit vectors3
UNIT VECTORS

Example 6

  • The given vector has length

    • So, by Equation 4, the unit vector with the same direction is:


Applications
APPLICATIONS

  • Vectors are useful in many aspects of physics and engineering.

    • In Chapter 13, we will see how they describe the velocity and acceleration of objects moving in space.

    • Here, we look at forces.


Force
FORCE

  • A force is represented by a vector because it has both a magnitude (measured in pounds or newtons) and a direction.

    • If several forces are acting on an object, the resultant forceexperienced by the object is the vector sum of these forces.


Force1
FORCE

Example 7

  • A 100-lb weight hangs from two wires.

  • Find the tensions (forces) T1 and T2in both wires and their magnitudes


Force2
FORCE

Example 7

  • First, we express T1 and T2in terms of their horizontal and vertical components.


Force3
FORCE

E. g. 7—Eqns. 5 & 6

  • From the figure, we see that: T1 = –|T1| cos 50° i + |T1| sin 50° j T2 = |T2| cos 32° i + |T2| sin 32° j


Force4
FORCE

Example 7

  • The resultant T1 + T2 of the tensions counterbalances the weight w.

  • So, we must have: T1 + T2 = –w = 100 j


Force5
FORCE

Example 7

  • Thus, (–|T1| cos 50° + |T2| cos 32°) i + (|T1| sin 50° + |T2| sin 32°) j = 100 j


Force6
FORCE

Example 7

  • Equating components, we get:

  • –|T1| cos 50° + |T2| cos 32° = 0

  • |T1| sin 50° + |T2| sin 32° = 100


Force7
FORCE

Example 7

  • Solving the first of these equations for |T2| and substituting into the second, we get:


Force8
FORCE

Example 7

  • So, the magnitudes of the tensions are:


Force9
FORCE

Example 7

  • Substituting these values in Equations 5 and 6, we obtain the tension vectors T1≈ –55.05 i + 65.60 j T2≈ 55.05 i + 34.40 j


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