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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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Vectors and the geometry of space

10

VECTORS AND

THE GEOMETRY OF SPACE


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.


Vectors and the geometry of space

VECTOR

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

DENOTING A VECTOR

  • We denote a vector by either:

    • Printing a letter in boldface (v)

    • Putting an arrow above the letter ( )


Vectors and the geometry of space

VECTORS

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


Vectors and the geometry of space

VECTORS

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

    • We indicate this by writing v = .


Vectors and the geometry of space

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.


Vectors and the geometry of space

ZERO VECTOR

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

    • It is the only vector with no specific direction.


Vectors and the geometry of space

COMBINING VECTORS

  • Suppose a particle moves from A to B.

  • So,its displacement vector is .


Vectors and the geometry of space

COMBINING VECTORS

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


Vectors and the geometry of space

COMBINING VECTORS

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


Vectors and the geometry of space

COMBINING VECTORS

  • The resulting displacement vector is called the sumof and .

  • We write:


Vectors and the geometry of space

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.


Vectors and the geometry of space

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.


Vectors and the geometry of space

VECTOR ADDITION

  • The definition of vector addition is illustrated here.


Vectors and the geometry of space

TRIANGLE LAW

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

VECTOR ADDITION

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

PARALLELOGRAM LAW

  • This is called the Parallelogram Law.


Vectors and the geometry of space

VECTOR ADDITION

Example 1

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

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.


Vectors and the geometry of space

VECTOR ADDITION

Example 1

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


Vectors and the geometry of space

MULTIPLYING VECTORS

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


Vectors and the geometry of space

SCALAR

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

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.


Vectors and the geometry of space

SCALAR MULTIPLICATION

  • The definition is illustrated here.

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

    • That’s why we call them scalars.


Vectors and the geometry of space

SCALAR MULTIPLICATION

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

SUBTRACTING VECTORS

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

SUBTRACTING VECTORS

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


Vectors and the geometry of space

SUBTRACTING VECTORS

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


Vectors and the geometry of space

SUBTRACTING VECTORS

Example 2

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

SUBTRACTING VECTORS

Example 2

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


Vectors and the geometry of space

COMPONENTS

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


Vectors and the geometry of space

COMPONENTS

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

COMPONENTS

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

COMPONENTS

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


Vectors and the geometry of space

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.


Vectors and the geometry of space

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|


Algebraic vectors4

ALGEBRAIC VECTORS

Example 4


Algebraic vectors5

ALGEBRAIC VECTORS

Example 4


Algebraic vectors6

ALGEBRAIC VECTORS

Example 4


Algebraic vectors7

ALGEBRAIC VECTORS

Example 4


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 vectors3

STANDARD BASIS VECTORS

Equation 2


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 jT2≈ 55.05 i + 34.40 j


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