Download +Chapter 8 Vectors and Parametric Equations 8.1/8.2 Geometric

+Chapter 8 Vectors and Parametric Equations
8.1/8.2 Geometric Vectors/Algebraic Vectors
Parallel:
Equal:
Opposite:
Adding Vectors
2 Ways to Represent Vectors
Finding Magnitude
Ex 1: Write the ordered pair that represents the vector from 𝑋(−3,5) to 𝑌(4, −2). Then find the magnitude
of ⃑⃑⃑⃑⃑
𝑋𝑌.
⃑⃑⃑ = ⟨5, −7⟩, 𝒏
⃑⃑ = ⟨0, 4⟩ and 𝒑
⃑ = ⟨−1, 3⟩. Find each of the following.
Ex 2: Let 𝒎
⃑⃑⃑ + 𝒑
⃑
a. 𝒎
⃑⃑⃑ − 𝒏
⃑
b. 𝒎
⃑⃑⃑ + 3𝒏
⃑⃑ − 𝒑
⃑
c. 2𝒎
Ex 3: Find the magnitude of each vector. Then write each vector as the sum of unit vectors.
a. ⟨3, 4⟩
b. ⟨−4, 1⟩
Finding Vertical and Horizontal Components
Ex 4: Find the vertical and horizontal components of a vector whose magnitude is 5 and direction is 96°.
8.5 Applications with Vectors
Ex 1: Find the magnitude and the direction of the resultant force to the nearest hundredth.
Ex 2: There are 3 forces acting on an object. The 1st force has the magnitude of 7 lbs. and is 90°. The 2nd
force has the magnitude of 5 lbs. and is 38° from force 1. The 3rd force has a magnitude of 11 lbs. and is 48°
below the positive x-axis. Find the resultant force and the direction.
Directions with a Compass
Ex 3: A boat travels 25 mph at a direction of 28° S of E. The water current is pushing on the boat with a speed
of 3 mph at a direction of 2° N of E. Find the actual speed and the direction of the boat.
8.3 Vectors in 3D
⃑ = ⟨−1, 2, 3⟩
Ex 1: Graph the ordered triple 𝒂
Ex 2: Let 𝑄(−37.5,281, −12) and 𝑅(−11, −12, 13)
⃑⃑⃑⃑⃑ as a sum of unit vectors
a. Write 𝑅𝑄
⃑⃑⃑⃑⃑
b. Find the magnitude of 𝑅𝑄
8.4 Perpendicular Vectors
Inner Product (Dot Product)
 Ex:
If the dot product = __________, then the vectors are _______________________.
Ex 1: Find the inner product of each set of vectors and state whether or not they are perpendicular.
2
⃑ = ⟨4,−12⟩
⃑ = ⟨5, ⟩ , 𝒃
a. 𝒂
3
⃑ = ⟨15, 4,−7⟩, ⃑𝒃 = ⟨5, −8, 9⟩
b. 𝒂
Matrices: Finding a determinant → Need a ________, _________, ________, etc.
Ex 2: Find the determinant of the following matrices
1 2
a. [
]
3 4
5 7
b. [−8 1
3 8
9
2]
0
−1 2
c. [ 0 4
−5 9
3
6]
1
Cross Product for 3D Vectors
The cross product of two vectors is a ____________________. This vector is _________________________ to
both vectors.
Ex:
⃑ if 𝒂
⃑ = ⟨7, 3, 2⟩
⃑ × 𝒃
⃑ = ⟨5, 6, 7⟩, 𝒃
Ex 3: Find 𝒂
Ex 4: Find a vector that is perpendicular to the plane containing 𝐴(5, −3,1), 𝐵(7, −2,6), 𝐶(3,0, −1)
8.6 Vectors and Parametric Equations
Vector Form of a Line
⃑ = ⟨𝑎1 ,𝑎2 ⟩ is defined by the set of
A line through 𝑃1 (𝑥1 , 𝑦1 ) ________________________ to the vector 𝒂
⃑ for some real number 𝑡.
points 𝑃1 (𝑥1 , 𝑦1 ) and 𝑃2 (𝑥2 , 𝑦2 ) such that ⃑⃑⃑⃑⃑⃑⃑⃑
𝑃1 𝑃2 = 𝑡𝒂
Therefore,
Parametric Form of a Line
The vector equation __________________________ written as two equations ________________________
Ex 1:
⃑ = ⟨3, −4⟩.
a. Write the equation of a line in vector form that passes through (2, −1) and is parallel to 𝒗
b. Write the parametric form of part a.
⃑ = ⟨−51, 43⟩ and passes through the
Ex 2: Write the vector AND parametric forms of a line that is parallel to 𝒂
point (12, −14).
5
Ex 3: Write parametric equations of 𝑦 = − 3 𝑥 − 5
Ex 4: Write parametric equations of 𝑦 = −4𝑥 + 7
Ex 5: Rewrite the equations from parametric form to standard form
𝑥 = −3𝑡 + 1
𝑦 = 5𝑡 − 8
Ex 6: Write an equation in slope intercept form of the line whose parametric equations are
𝑥 = −5 + 4𝑡
𝑦 = 2 − 3𝑡
Ex 7: Graph the equation. Indicate the direction.
a. 𝑥 = 3𝑡 − 5
𝑦 = −2𝑡 + 1
b. 𝑥 = −2𝑡 − 3
𝑦 = 4𝑡 − 1
8.7 Modeling Projectile Motion
Parametric Equations for Projectile Motion
Horizontal Distance:
Vertical Distance:
Ex 1: John is kicking a soccer ball from the ground with initial velocity of 45 ft/sec at an angle of 11° with the
horizontal.
a. Write parametric equations to model the motion
b. How long is the ball airborne?
c. At what time was the ball at it’s highest? What is the maximum height?
Ex 2: Brady kicks a football with an initial velocity of 29 yd/sec at an angle of 68° with the horizontal. Find the
range of the ball and the maximum height to the nearest hundredth.
Ex 3: Oprah pitches a softball 2.9 ft above the ground with a speed of 85 mph at an angle of 5.1° with the
horizontal. The distance between Oprah and home plate is 60.5 ft.
a. How long is the ball airborne?
b. When the ball passes home plate, what is its height?