Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

Objective:

VECTORS

• Use magnitude and direction to show vectors are equal.

• Visualize scalar multiplication, vector addition, and vector subtraction as geometric vectors.

• Represent vectors in the rectangular coordinate system.

• Perform operations with vectors in terms of i and j.

• Find the unit vector in the direction of v.

• Write a vector in terms of its magnitude and direction.

• Solve applied problems involving vectors.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

Vector

VECTORS

Ours is a world of pushes and pulls. For example, suppose you are pulling a cart up a 30° incline, requiring an effort of 100 pounds. This quantity is described by giving its magnitude (a number indicating size, including a unit of measure) and also its direction.

The magnitude is 100 pounds and the direction is 30° from the horizontal. Quantities that involve both a magnitude and a direction are called vector quantities, or vectors for short.

Here is another example of a vector:

You are driving due north at 50 miles per hour.

The magnitude is the speed, 50 miles per hour. The direction of motion is due north.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Some quantities can be completely described by giving only their magnitudes. For example, the temperature of the lecture room that you just left is 75°. This temperature has magnitude, 75°, but no direction. Quantities that involve magnitude, but no direction, are called scalar quantities, or scalars for short. Thus, a scalar has only a numerical value. Another example of a scalar is your professor’s height, which you estimate to be 5.5 feet.

In this section and the next, we introduce the world of vectors, which literally surround your every move. Because vectors have nonnegative magnitude as well as direction, we begin our discussion with directed line segments.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

Vector

VECTORS

Quantities that involve both a magnitude and a direction are called vector quantities, or vectors for short.

Quantities that involve magnitude, but no direction, are called scalar quantities, or scalars for short.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Directed Line Segments and Geometric VectorsA line segment to which a direction has been assigned is called a directed line segment. We call the initial point and the terminal point. We denote this directed line segment by .

The magnitude of the directed linesegment is its length. We denote this by .Geometrically, a vector is a directedline segment.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Representing Vectors in Print and on PaperVectors are often denoted by boldface letters, such as v. If a vector v has the same magnitude and the same direction as the directed line segment we write .

𝒗= �⃗�𝑸

Because it is difficult to write boldface on paper, use an arrow over a single letter, such as to denote v, the vector v.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Representing Vectors in Print and on Paper

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

In general, vectors v and w are equal if they have the same magnitude and the same direction. We write this as v = w.

Equal Vectors

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Show that

Equal vectors have the same magnitude and the same direction. Use the distance formula to show that u and v have the same magnitude.

‖𝒖‖=√( 𝒙𝟐− 𝒙𝟏 )𝟐+ (𝒚𝟐− 𝒚𝟏 )𝟐

¿√ (𝟔−𝟐 )𝟐+(−𝟐− (−𝟓 ) )𝟐

¿√ (𝟒 )𝟐+ (𝟑 )𝟐

¿√𝟏𝟔+𝟗¿√𝟐𝟓¿𝟓

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Show that

Equal vectors have the same magnitude and the same direction. Use the distance formula to show that u and v have the same magnitude.

‖𝒗‖=√( 𝒙𝟐−𝒙𝟏)𝟐+ (𝒚𝟐− 𝒚𝟏 )𝟐

¿√ (𝟔−𝟐 )𝟐+(𝟓−𝟐 )𝟐

¿√ (𝟒 )𝟐+ (𝟑 )𝟐

¿√𝟏𝟔+𝟗¿√𝟐𝟓¿𝟓

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Show that

One way to show that u and v have the same direction is to find the slopes of the lines on which they lie.

𝒎=𝒙𝟐−𝒙𝟏

𝒚𝟐−𝒚𝟏

¿𝟔−𝟐

−𝟐− (−𝟓 )¿𝟒𝟑

𝒎=𝒙𝟐−𝒙𝟏

𝒚𝟐−𝒚𝟏

¿𝟔−𝟐𝟓−𝟐

¿𝟒𝟑

slope of u

slope of v

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Show that

‖𝒖‖=𝟓 ‖𝒗‖=𝟓

𝑺𝒍𝒐𝒑𝒆𝒐𝒇 𝒖=𝟒𝟑

𝑺𝒍𝒐𝒑𝒆𝒐𝒇 𝒗=𝟒𝟑

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Scalar MultiplicationThe multiplication of a real number k and a vector v is called scalar multiplication. We write this product kv.

Multiplying a vectorby any positive realnumber (except 1)changes the magnitudeof the vector but not its direction.

Multiplying a vector by any negative number reverses the direction of the vector.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Scalar Multiplication

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

The Sum of Two VectorsThe sum of u and v, denoted u + v is called the resultant vector. A geometric method for adding two vectors is shown in the figure. Here is how we find this vector:

• Position u and v, so that the terminal point of u coincides with the initial point of v.

• The resultant vector, u + v, extends from the initial point of u to the terminal point of v.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

The Difference of Two VectorsThe difference of two vectors, v – u, is defined as v – u = v + (–u), where –u is the scalar multiplication of u and –1, –1u. The difference v – u is shown geometrically in the figure.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Vectors in the Rectangular Coordinate System

As you saw in Example 1, vectors can be shown in the rectangular coordinate system.

Now let’s see how we can use the rectangular coordinate system to represent vectors. We begin with two vectors that both have a magnitude of 1. Such vectors are called unit vectors.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

The i and j Unit Vectors

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Representing Vectors in Rectangular CoordinatesWhy are the unit vectors i and j important? Vectors in the rectangular coordinate system can be represented in terms of i and j.

For example, consider vector v with initial point at the origin, , and terminal point at We can represent v using i and j as .

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Representing Vectors in Rectangular Coordinates

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Sketch the vector v = 3i – 3j and find its magnitude.

initial point

terminal point

v = 3i –

3j

𝒗=𝒂𝒊+𝒗𝒋𝒗=𝟑 𝒊−𝟑 𝒋

𝒂=𝟑𝒃=−𝟑 ‖𝒗‖=√( 𝒙𝟐−𝒙𝟏)𝟐+ (𝒚𝟐− 𝒚𝟏 )𝟐

¿√ (𝟑 )𝟐+(−𝟑 )𝟐

¿√𝟗+𝟗¿√𝟏𝟖¿𝟑√𝟐

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Representing Vectors in Rectangular Coordinates

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Let v be the vector from initial point P1 = (–1, 3) to terminal point P2 = (2, 7). Write v in terms of i and j.

v = 3i + 4j

𝒗=( 𝒙𝟐−𝒙𝟏 ) 𝒊+(𝒚 𝟐−𝒚𝟏 ) 𝒋

¿ [𝟐− (−𝟏 ) ] 𝒊+(𝟕−𝟑 ) 𝒋

¿𝟑 𝒊+𝟒 𝒋𝑷 𝟏=(−𝟏 ,𝟑 )

𝑷 𝟐=(𝟐 ,𝟕 )

(𝟑 ,𝟒)

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Adding and Subtracting Vectors in Terms of i and j

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

If v = 7i + 3j and w = 4i – 5j, find the following vectors:a) v + w

b) v – w

1 2 1 2( ) ( )v w a a i b b j

(7 4) [3 ( 5)]i j

11 2i j

1 2 1 2( ) ( )v w a a i b b j

(7 4) [3 ( 5)]i j

3 8i j

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Scalar Multiplication with a Vector in Terms of i and j

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

If v = 7i + 10j, find each of the following vectors:a. 8v

b. –5v

( ) ( )kv ka i kb j 8 (8 7) (8 10)v i j

56 80i j

( ) ( )kv ka i kb j

5 ( 5 7) ( 5 10)v i j 35 50i j

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

The Zero Vector

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Properties of Vector Addition

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Properties of Vector Addition

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Unit VectorsA unit vector is defined to be a vector whose magnitude is one.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Find the unit vector in the same direction as v = 4i – 3j. Then verify that the vector has magnitude 1.

2 2v a b 2 24 ( 3) 16 9 25 5

4 35

v i jv

2 24 35 5

4 35 5i j

16 925 25

251

25

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Writing a Vector in Terms of Its Magnitude and Direction

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

The jet stream is blowing at 60 miles per hour in the direction N45°E. Express its velocity as a vector v in terms of i and j.

v

45 , 60v cos sinv v i v j

60cos45 60sin 45v i j 2 2

60 602 2i j

30 2 30 2i j

The jet stream can be expressed in terms of i and j as

30 2 30 2i j

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

ApplicationMany physical concepts can be represented by vectors. A vector that represents a pull or push of some type is called a force vector. If you are holding a 10-pound package, two force vectors are involved. The force of gravity is exerting a force of magnitude 10 pounds directly downward. This force is shown by vector in the figure. Assuming there is no upward or downward movement of the package, you are exerting a force of magnitude 10 pounds directly upward. This force is shown by vector in Figure 6.61.

It has the same magnitude as the force exerted on your package by gravity, but it acts in the opposite direction. If and are two forces acting on an object, the net effect is the same as if just the resultant force, acted on the object. If the object is not moving, as is the case with your 10-pound package, the vector sum of all forces is the zero vector.

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Two forces, F1 and F2, of magnitude 30 and 60 pounds, respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force.

x

y Resultantforce, F

F2

60 pounds

F1

30 pounds

10N E

60N E

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

x

y Resultantforce, F

F2

60 pounds

F1

30 pounds

10N E

60N E

1 1 1cos sinF F i F j 30cos80 30sin80i j 5.21 29.54i j

2 2 2cos sinF F i F j 60cos30 60sin30i j 51.96 30i j

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

1 5.21 29.54F i j

2 51.96 30F i j

1 2F F F (5.21 29.54 ) (51.96 30 )i j i j (5.21 51.96) (29.54 30)i j 57.17 59.54i j

2 2F a b 2 257.17 59.54 82.54

cosaF

57.1782.54

1 57.17

cos82.54

46.2

Mrs. Rivas

International Studies Charter

School.

Pre-Calculus

Section 6-6

VECTORS

Two forces, F1 and F2, of magnitude 30 and 60 pounds, respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force.

The two given forces are equivalent to a single force of approximately 82.54 pounds with a direction angle of approximately 46.2°.

x

y Resultantforce, F

F2

60 pounds

F1

30 pounds

10N E

60N E

Mrs. RivasHomework

Pg. 709-710 # 2, 4, 8, 12, 14, 18, 20, 28, 32, 38, 42, 46, 48, 52, 54, 56, 58, 62, 72, 74