accelerated precalculus name introduction to vectors date ... · the vector sum + is called a...

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Accelerated Precalculus Name_____________________________ Introduction to Vectors Date____________________ Block_____

What is a vector?

A vector is a quantity that has both magnitude and direction.

State whether each quantity described is a vector quantity or a scalar quantity.

a. A boat traveling at 15 miles per hour.

b. A hiker walking 25 paces due west

c. A person’s weight on a bathroom scale

Component Form of a Vector

The component form of a vector AB with initial point A(x1,y

1) and terminal point B(x

2,y

2) is given by

⟨𝑥2 − 𝑥1, 𝑦2 − 𝑦1⟩.

Example: 1. Find the component form of vector AB with initial point A(-4,2) and terminal point B(3,-5). 2. Find the component form of vector AB with initial point A(-2,-7) and terminal point B(6,1).

Magnitude of a Vector

If v is a vector with initial point (x1, y

1) and terminal point (x

2, y

2), then the magnitude of v is given by

2 2

2 1 2 1( ) ( )v or v x x y y= − + −

If v has a component form of ⟨𝑎, 𝑏⟩ then 2 2v or v ba= +

Examples

Let v be a vector with initial point A and terminal point B. Find the magnitude of v, with the given initial and terminal points

3. A(-4,2), B(3,-5)

4. A(-2,-7), B(6,1)

5. ⟨𝟑, −𝟐⟩

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Direction Angle of a Vector

The direction angle, 𝜃, of a vector can be found by using tangent. tan =b

a.

So if 10; tan− =

ba

aand if 10; tan 180−

= +

b

aa

Examples

Find the direction angle of each vector to the nearest tenth of a degree. 6. 𝑝 = ⟨7, −3⟩

7. 𝑟 = ⟨4, −5⟩

8. 𝑠 = ⟨−4, 2⟩

9. 𝑟 = ⟨−6, −4⟩

All Together Now

10. Given A(−2,4) and B(3,2), find the component form, magnitude, and direction of AB . Sketch the vector and label

those 3 characteristics.

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Vector Operations

If 𝑎 = ⟨𝑎1, 𝑎2⟩ and 𝑏 = ⟨𝑏1, 𝑏2⟩ are vectors, then the following are true.

Vector Addition 𝑎 + 𝑏 = ⟨𝑎1 + 𝑏1, 𝑎2 + 𝑏2⟩

Vector Subtraction 𝑎 − 𝑏 = ⟨𝑎1 − 𝑏1, 𝑎2 − 𝑏2⟩

Scalar multiplication 𝑘𝑎 = ⟨𝑘𝑎1, 𝑘𝑎2⟩

Examples

Complete the following operations given

11. +u v 12. 6−u v

13. 5 2−v u 14. 2 3+u v

Linear Combination

The unit vectors in the direction of the positive x-axis and positive y-axis are denoted by

i = ⟨1,0⟩ and j = ⟨0,1⟩. Vectors i and j are called standard unit vectors.

The vector sum 𝑎𝑖 + 𝑏𝑗 is called a linear combination of the vectors 𝑖 and j.

Example

15. Let DE be the vector with initial point D(-2,3) and terminal point E(4,5). Write DE as a linear combination of the vectors 𝑖 and j.

7,2 and=u 3,5 .= −v

11. 5 2−v u

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Introduction to Vectors Practice WS

State whether each quantity described is a vector quantity or a scalar quantity.

1. A box being pushed with a force of 125 newton’s

2. Wind blowing at 20 knots

3. A deer running 15 meters per second due west

4. A baseball thrown with a speed of 85 miles per hour

5. A 15-pound tire hanging from a rope

6. A rock thrown straight up at a velocity of 50 feet per second

Find the component form of AB . Then find the magnitude and direction of AB .

7. 𝑨(2,3), 𝑩(5, −7)

8. 𝑨(3,0), 𝑩(−4, −5)

9. 𝑨(−2, 5), 𝑩(7, −3)

10. 𝑨(−2, −1), 𝑩(6,6)

11. 𝑨(−2, −4), 𝑩(6, −4)

12. 𝑨(5, −2), 𝑩(1,1)

13. 𝐴(6, 0), 𝐵(5, 5)

14. 𝑨(1,2), 𝑩(2, 1)

15. 𝑨(3, −4), 𝑩(−9, −8)

16. 𝑨(3, 1), 𝑩(6, −3)

Perform the indicated operations

𝑨 = ⟨3,6⟩ 𝑩 = ⟨−4, 9⟩ 𝑪 = ⟨2, −8⟩ 𝑫 = ⟨−5, −4⟩ 𝑬 = ⟨0, −2⟩

17. 2𝐴 − 𝐵

18. 𝐷 − 3𝐶

19. 5𝐴 + 𝐶 − 2𝐷

20. 𝐸 − 𝐷 − 𝐶

21. 5𝐴 + 2𝐷 − 𝐸

22. 4𝐴 + 𝐷 − 𝐵

23. 𝐶 − 𝐶 − 2𝐵

24. 𝐴 − 𝐸 + 𝐵

25. 3𝐵 − 𝐶

26. 𝐴 + 𝐵 + 𝐶 + 𝐷

27. 𝐴 − 𝐵 − 𝐷 − 𝐶

28. 𝐸 − 𝐵 + 2𝐴

29. 𝐴 + 4𝐵 − 𝐶 30. 2𝐴 + 3𝐵 + 4𝐶 − 5𝐷 − 6𝐸

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Accelerated Precalculus Dot Products and Resultants

Let 1 2,a a a= .

Then 1 2 1 2, ,a a a a a− =− = − −

Example

*There is a second method called the parallelogram method, but I don’t care for it and do not use it. See

completed notes.

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x

y

x

y

x

y

x

y

Graph each of the following on the given plane. Make sure you show your Resultant Vector.

1. 𝟐𝒃 + 𝒅

2. 𝒂 − 𝒆

3.

𝒄 + 𝟑𝒅

4. 𝟐𝒅 − 𝟐𝒃

5.

𝒄 − 𝒂

6. 𝟐𝒂 − 𝟑𝒃 + 𝒅 − 𝒆

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x

y

x

y

Dot Products and Resultants Additional Practice Find the dot product of u and v.

1. 𝑢 = ⟨3, −5⟩, 𝑣 = ⟨6,2⟩ 2. 𝑢 = ⟨−10, −16⟩, 𝑣 = ⟨−8,5⟩

3. 𝑢 = ⟨9, −3⟩, 𝑣 = ⟨1,3⟩ 4. 𝑢 = ⟨4, −4⟩, 𝑣 = ⟨7,5⟩

5. 𝑢 = ⟨1, −4⟩, 𝑣 = ⟨2,8⟩ 6. 𝑢 = 11𝑖 + 7𝑗; 𝑣 = −7𝑖 + 11𝑗

7. 𝑢 = ⟨−4,6⟩, 𝑣 = ⟨−5, −2⟩ 8. 𝑢 = 8𝑖 + 6𝑗; 𝑣 = −𝑖 + 2𝑗

Graph each of the following on the given plane. Make sure you show your Resultant Vector.

9. 2𝑇 + 𝑅

10. 𝑈 − 𝑆

V

S

U

R

T

8

x

y

x

y

x

y

x

y

11. 2𝑉 + 2𝑆

12. 𝑈 + 𝑇

13. 3𝑆 + 𝑅 − 𝑉

14. 𝑈 + 2𝑇 − 𝑆

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Vector Operations Review Worksheet Name __________________________________________

Using the vectors below, find the following:

𝑨 = ⟨−4,1⟩ 𝑩 = ⟨3,6⟩ 𝑪 = ⟨1, −5⟩ 𝑫 = ⟨−2, −6⟩

1. 𝑨 + 𝑩

2. 2𝑨 − 4𝑪

3. 𝑪 − 𝑫

4. 3𝑪 − 5𝑩

5. 𝑨 − 𝑫 + 𝑪 6. ‖𝑨‖ + ‖𝑩‖

7. ‖𝑪‖ − ‖𝑫‖ 8. 𝑪 ∙ 𝑨

9. 𝑩 ∙ 𝑫 10. 2(𝑨 + 𝑫) − 4𝑩

11. 𝑩 ∙ 𝑩 12. ‖𝑪‖ ∙ ‖𝑪‖

13. What is the direction angle for vector A? 14. What is the direction angle for vector B?

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15. What is the direction angle for vector C? 16. What is the direction angle for vector D?

17. Graph vectors A and B (be sure to label the vectors)

18. Graph vectors C and D (be sure to label the vectors)

Find the component form of AB . Then find the magnitude and direction of AB .

19. 𝐴(−3, −4), 𝐵(2,1)

20. 𝐴(3,6), 𝐵(2, −2)

21. 𝐴(4,1), 𝐵(−3,2)

22. 𝐴(2,4), 𝐵(−3, −6)

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Let 𝒗 = ⟨2, −1⟩ 𝑎𝑛𝑑 𝒘 = ⟨−3,5⟩. Find 𝑢 algebraically and sketch the vector operations geometrically.

23. 𝒖 = 𝒗 + 𝒘

24. 𝒖 = 𝒗 − 𝒘

25. 𝒖 = 3𝒗

26. 𝒖 = 𝒘 − 2𝒗

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Graph each of the following on the given plane. Make sure you show your resultant vector.

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More on Vector Operations

Linear Combination

The unit vectors in the direction of the positive x-axis and positive y-axis are denoted by

i = ⟨1,0⟩ and j = ⟨0,1⟩. Vectors i and j are called standard unit vectors.

The vector sum 𝑎𝑖 + 𝑏𝑗 is called a linear combination of the vectors 𝑖 and j.

Write the component form vector as a standard unit vector. Validate your answer (show proof)!

3. 7,2 4. 4,6 −

Find the direction angle of each vector.

5. 2 3i j− 6. 4 5i j− +

7. Find component form of the vector v with magnitude 10 and directions angle 120o.

Find component form of a vector with the given magnitude and direction.

11. 5 2−v u

1.

.

2.

.

8.

.

9.

.

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Finding the Angle Between Vectors

Use the Law of Cosines a formula to find the angle between the given vectors. Assume that 0 180 .

9. 2 5 , 6 2v i j w i j= − + = +

10. 3 2 , 4v i j w i j= + = −

A bit more vocabulary and extension!

11. Given a vector with initial point ( )2, 5− and terminal point ( )7,4− , find the position vector.

This is referring to a vector in standard position, which we also call a vector written in component form.

12. If 2 vectors have a dot product that equals zero, then those vectors are said to be orthogonal. Orthogonal means the vectors are perpendicular to each other.

Determine if the given vectors are orthogonal.

a.) 3,7= −u , 5,2=v b.) 2,5=u , 10, 4= −v