Math Textbook: 6.3 Vectors in the Plane

The negative of v=(v₁,v₂) is...?

-v=(-1)v =(-v₁,-v₂)

Writing a Linear Combination of Unit Vectors: Let u be the vector with initial point (2,-5) and terminal point (-1,3). Write u as a linear combination of the standard unit vectors i and j.

1. Begin by writing the component form of the vector u u=(-1-2,3-(-5)) =(-3,8) =-3i+8j

Finding the Component Form of a Vector: Find the component form of the vector that represents the velocity of an airplane descending at a speed of 100 miles per hour at an angle of 30° below the horizontal, as shown in Figure 6.22.

1. The velocity vector v has a magnitude of 100 and a direction angle of θ=210° v=‖v‖(cosθ)i+‖v‖(sinθ)j v=100(cos210)i+100(sin210)j v=100(-√3/2)i+100(-1/2)j =-50i√3-50j =(-50√3, -50)

In many applications of vectors, it is useful to find a...?

A unit vector that has the same direction as a given nonzero vector v.

The set of all directed line segments that are equivalent to a given directed line segment →PQ is a...?

A vector v in the plane.

Any vector in the plane can be written as a ....?

As a linear combination of the standard unit vectors i and j

Vector Operations: Let v=(-2,5) and w=(3,4). Find 2v

Because v=(-2,5), you have 2v=2(-2,5) =(2(-2), 2(5)) =(-4,10)

The standard unit vectors can be used to.....?

Can be used to represent any vector v=(v₁,v₂) as follows: v=(v₁,v₂) =v₁(1,0)+v₂(0,1) =v₁i+v₂j

The angle θ is the _______ of the vector u.

Direction angle

To add two vectors u and v geometrically, what do you first do?

First position them (without changing their lengths or directions) so that the initial point of the second vector v coincides with the terminal point of the first vector u

The scalars v₁ and v₂ are called the....?

Horizontal and vertical components of v

‖v‖=0 IF and ONLY IF...?

If and only if v is the zero vector 0

What does the directed line segment →PQ have?

Initial point P and terminal point Q.

What do both force and velocity involve? What can they cannot be completely characterized by?

Involve both magnitude and direction and cannot be completely characterized by a single real number.

The vector u+v is often called the...?

Resultant

Vector Operations: Let v=(-2,5) and w=(3,4). Find w-v

The difference of w and v is w-v=(3,4)-(-2,5) =(3-(-2), 4-5) =(5,-1)

If k is negative, then kv has...?

The opposite direction of v

If k is positive, then kv has...?

The same direction as v

Vector Operations: Let v=(-2,5) and w=(3,4). Find v+2w

The sum of v and 2w is v+2w=(-2,5)+2(3,4) =(-2,5)+(6,8) =(4,13)

If u is a unit vector such that θ is the angle from the positive x-axis to u, then the terminal point of u ______ and you have _______

The terminal point of u lies on the unit circle and you have: u=(x,y) u=(cosθ,sinθ) u=(cosθ)i+(sinθ)j

The difference u-v is the vector...?

The vector from the terminal point of v to the terminal point of u, which is equal to u+(-v)

Geometrically, the product of a vector v and a scalar k is the vector that is...?

The vector that is |k| times as long as v

If ‖v‖=1, then v...?

Then v is a unit vector.

If both the initial point and the terminal point lie at the origin, then v is ____ and is denoted by _____?

Then v is the zero vector and is denoted by 0=(0,0)

U is a ____ of v.

U is a scalar multiple of v.

How are vectors denoted?

Vectors are denoted by lowercase, boldface letters such as u, v, and w

What is the Distance Formula?

d=√(x₂-x₁)²+(y₂-y₁)²

What are the standard unit vectors denoted by?

i=(1,0) and j=(0,1)

Let u=(u₁,u₂) and v=(v₁,v₂) be vectors and let k be a scalar (a real number). Then the scalar multiple of k times u is the vector...?

ku =k(u₁,u₂) =(ku₁,ku₂)

Let u=(u₁,u₂) and v=(v₁,v₂) be vectors and let k be a scalar (a real number). Then the sum of u and v is the vector...?

u+v=(u₁+v₁ , u₂+v₂)

The difference of u and v is...?

u-v =u+(-v) =(u₁-v₁,u₂-v₂)

Two vectors u=(u₁,u₂) and v=(v₁,v₂) are equal IF and ONLY IF...?

u₁=v₁ and u₂=v₂

How is a vector in the plane written?

v=→PQ

The magnitude (or length) of v is given by...?

‖v‖ =√(q₁-p₁)²+(q₂-p₂)² =√v₁²+v₂²

The component form of the vector with initial point P(p₁,p₂) and terminal point Q(q₁,q₂) is given by...?

→PQ =(q₁-p₁,q₂-p₂) =(v₁,v₂) =v

Finding Direction Angles of Vectors: Find the direction angle of vector u=3i+3j

1. The direction angle is determined from tanθ=b/a tanθ=b/a tanθ=3/3 tanθ=1 2. So, θ=45°

Finding a Unit Vector: Find a unit vector in the direction of v=(-2,5) and verify that the result has a magnitude of 1.

1. The unit vector in the direction of vi is v/‖v‖ =(-2,5)/√(-2)²+5² =(-2,5)*1/√4+25 =(-2,5)*1/√29 =(-2/√29, 5/√29) =(-2√29/29, 5√29/29) 2. This vector has a magnitude of 1 because √(-2√29/29)²+(5√29/29)² =√116/841+725/841 =√841/841 =1

Let u, v, and w be vectors and let c and d be scalars. Finish the 9 properties. 1. u+v=? 2. (u+v)+w=? 3. u+0=? 4. u+(-u)=? 5. c(du)=? 6. (c+d)u=? 7. c(u+v)=? 8. 1(u)=?, 0(u)=? 9. ‖cv‖=?

1. u+v=v+u 2. (u+v)+w=u+(v+w) 3. u+0=u 4. u+(-u)=0 5. c(du)=(cd)u 6. (c+d)u=cu+du 7. c(u+v)=cu+cv 8. 1(u)=u, 0(u)=0 9. ‖cv‖=|c|‖v‖

What does it mean to be in "standard position"?

A directed line segment whose initial point is the origin.

The vector sum v₁i+v₂j is called a....?

A linear combination of the vectors i and j

To represent u-v geometrically, you can use....?

Directed line segments with the SAME initial point

Two directed line segments that have the same magnitude and direction are....?

Equivalent.

Assume that u is a unit vector with the direction angle θ. If v=ai+bj is any vector that makes an angle θ with the positive x-axis, then ______ and you can write ______.

It has the same direction as u and you can write: v =‖v‖(cosθ,sinθ) =‖v‖(cosθ)i+‖v‖(sinθ)j

The sum u+v is the vector formed by...?

Joining the initial point of the first vector u with the terminal point of the second vector v

In operations with vectors, what are numbers usually referred to as?

Scalars

The unit vectors (1,0) and (0,1) are called the...?

Standard unit vectors

How do you find a unit vector that has the same direction as a given nonzero vector v?

You can divide v by its magnitude to obtain: u =unit vector =v/‖v‖ =v(1/‖v‖)

Using Vectors to Determine Weight: A force of 600 pounds is required to pull a boat and trailer up a ramp inclined at 15° from the horizontal. Find the combined weight of the boat and trailer. Figure 6.23

1. Based on figure 6.23, you can see: ‖BA‖=force of gravity=combined weight of boat and trailer ‖BC‖=force against ramp ‖AC‖=force required to move boat up ramp=600 pounds 2. By construction, triangles BWD and ABC are similar. So, angle ABC is 15°. In ∆ABC you have sin15=‖AC‖/‖BA‖ sin15=600/‖BA‖ ‖BA‖=600/sin15 ‖BA‖=2318 pounds 3. Combined weight is approximately 2318 pounds

Showing that Two Vectors Are Equivalent: Show that u and v are equivalent. u: (0,0) to (3,2) u: P to Q v: (1,2) to (4,4) v: R to S

1. From the Dist. Formula, it follows that →PQ and →RS have the same magnitude ‖→PQ‖=√(3-0)²+(2-0)² =√13 ‖→RS‖=√(4-1)²+(4-2)² =√13 2. Both line segments have the same direction, because they are both directed toward the upper right on lines having the same slope. Slope of →PQ=2-0/3-0 =2/3 Slope of →RS=4-2/4-1 =2/3 3. Because →PQ and →RS have the same magnitude and direction, u and v are equivalent.

Finding the Component Form of a Vector: Find the component form and magnitude of the vector v represented by the directed line segment that has initial point (4,-7) and terminal point (-1,5)

1. Let P(4,-7)=(p₁,p₂) and Q(-1,5)=(q₁,q₂) 2. Then, the components of v=(v₁,v₂) are v₁=q₁-p₁=-1-4=-5 v₂=q₂-p₂=5-(-7)=12 3. So, v=(-5,12) and the magnitude of v is ‖v‖=√(-5)²+(12)²=13

What are the two basic vector operations?

1. Scalar multiplication 2. Vector addition

Finding Direction Angles of Vectors: Find the direction angle of vector v=3i-4j

1. The direction angle is determined from tanθ=b/a tanθ=b/a tanθ=-4/3 2. Because v=3i-4j lies in Quadrant IV, θ lies in Quadrant IV and its reference angle is θ'=|arctan(-4/3)|=53.13° 3. So, it follows that θ=360-53.13 θ=306.87°

Using Vectors to Find Speed and Direction: An airplane is traveling at a speed of 500 miles per hour with a bearing of 330° at a fixed altitude with a negligible wind velocity, as shown in Figure 6.24a. (A bearing of 330° corresponds to a direction angle of 120°). The airplane encounters a wind blowing with a velocity of 70 miles per hour in the direction N45°E, as shown in figure 6.24b. What are the resultant speed and direction of the airplane?

1. Using figure 6.24, the velocity of the airplane alone is v₁=500(cos120,sin120) =(-250, 250√3) 2. And the velocity of the wind is v₂=70(cos45,sin45) =(35√2,35√2) 3. So, the velocity of the airplane in the wind is v=v₁+v₂ =(-250, 250√3)+(35√2,35√2) =(-200.5,482.5) 4. The resultant speed of the airplane is ‖v‖=√(-200.5)²+(482.5)² =522.5 miles per hour 5. To find θ, the direction <, tanθ=482.5/-200.5=-2.4065 6. The flight path lies in Quadrant II, so θ lies in Quadrant II and its reference angle is θ=|arctan(-2.4065)| θ=67.4° 7. So, the direction < is θ=180-67.4 =112.6°

How is the directed line segment →PQ's magnitude denoted? How can it be found?

Denoted by ‖→PQ‖ and can be found by using the Distance Formula.

What is a vector that can be uniquely represented by the coordinates of its terminal point? How can this be written?

The component form of a vector v, written as v=(v₁,v₂)

What are the coordinates v₁ and v₂?

The components of v.

A vector whose initial point is at the origin can be uniquely represented by...?

The coordinates of its terminal point (v₁,v₂)

The vector u has a magnitude of ____ and the _____ as v.

The vector u has a magnitude of 1 and the same direction as v.

The vector u is called a ________

The vector u is called a unit vector in the direction of v.

How do you represent such a quantity with magnitude and direction?

You can use a directed line segment. Has an arrow, pointed from one point to the other.

Vector Operations: Let u=-3i+8j and v=2i-j. Find 2u-3v.

You could solve this problem by converting u and v to component form, but this isn't necessary. Performing the operations in unit vector form gives the same results. 2u-3v =2(-3i+8j)-3(2i-j) =-6i+16j-6i+3j =-12i+19j