1. Lesson 12 – 7 Geometric Vectors
Pg. 643 #1, 5, 10, 13, 18, 20, 27, 29, 30, 32, 35–39
Lesson 12 – 7 Geometric Vectors
Pre-calculus
To draw vectors
To find the norm (magnitude) of vectors
To use graphs of vectors

2. Learning Objective To draw vectors
To find the norm (magnitude) of vectors
To use graphs of vectors

3. Vectors 𝐴𝐵 Vector: a quantity that has both magnitude and direction. B
head A
tail 𝐴𝐵
Vectors can be placed anywhere on a grid, not necessarily starting at the origin
Note: the textbook will use bold print v, we will write 𝑣
The magnitude or length or norm of a vector can be found using the Pythagorean Theorem 𝑣
***Vectors are equivalent if they have the same magnitude and direction no matter where they are located.

4. Vectors If you add two vectors, you get a resultant vector
There are two methods. Note: Subtracting vectors is like adding the opposite of the second vector. 𝑣
1. Find 𝑢 + 𝑣 𝑢
Method 1:
Tail–to–head
Method 2:
Parallelogram 𝑢
𝑢 + 𝑣 𝑣
𝑢 + 𝑣 𝑣 𝑣 𝑢 𝑢

5. Vectors 2. Find 𝑢 − 𝑣 = 𝑢 +(− 𝑣 ) 𝑣 𝑢 Method 1: Tail–to–head Method 2:
= 𝑢 +(− 𝑣 )
Vectors 𝑣 𝑢
Method 1:
Tail–to–head
Method 2:
Parallelogram 𝑢 𝑢
− 𝑣
𝑢 − 𝑣
− 𝑣
𝑢 − 𝑣
− 𝑣 𝑢

6. Zero Vector: If 𝑣 = 𝑢 , then 𝑣 − 𝑢 =0 is the zero vector (magnitude = 0)
Vectors
A scalar will change the length of a vector
If 𝑎 is a scalar and 𝑣 is a vector,
𝑎 𝑣 is longer if 𝑎 >1
𝑎 𝑣 is shorter if 𝑎 <1
* If 𝑎 is negative, the direction reverses

7. Vectors 3. Use vector 𝑣 to draw & find the norm of 0.5 𝑣 4 4 0.5 𝑣
= 𝑣
=0.5( )
=0.5( 25 )
=0.5(5)
=2.5

8. We can find the horizontal & vertical components of a vector using trig.
Vectors 𝑣
𝑦=𝑟 sin 𝜃 𝜃
𝑥=𝑟 cos 𝜃

9. 4. Given that vector 𝑤 has a magnitude of 10 and a direction of 135 𝑜 , find the 𝑥 and 𝑦 components.
Vectors
5 2 10
135°
−5 2
=10 −
𝑥=10 cos 135 𝑜
=−5 2 =
𝑦=10 sin 135 𝑜
=5 2

10. Vectors can be used together with the Law of Cosines or the Law of Sines to solve problems involving physical quantities such as force, velocity, and acceleration
Vectors N
heading
air velocity
course
wind
velocity
ground velocity
Norm of air velocity = air speed
Direction of air velocity = plane’s heading
Measured CLOCKWISE from the north
Ground velocity = vector sum of air velocity and wind velocity
Course = direction of ground velocity (angle from the north)

11. 5. Aviation: An airplane has an air speed of 520 mph and a heading of 115 𝑜 . Wind blows from the east at 37 mph. Find the plane’s ground speed (norm of ground velocity) and course.
Vectors N
115°
90°
25°
𝛼 𝑜
520 𝑣
25° 37
Find 𝑣 and 𝛼 𝑜
3 sides … Law of Cosines
𝑣 2 = − cos 25 𝑜
Plane’s Ground Speed = 𝑣 =487 𝑚𝑝ℎ
So course is 115 𝑜 + 2 𝑜
= 117 𝑜
sin 𝛼 𝑜 = sin 25 𝑜 487
𝛼= 𝑠𝑖𝑛 −1 ( 37sin 25 𝑜 )
𝛼= 2 𝑜

12. Assignment
Pg. 643 #1, 5, 10, 13, 18, 20, 27, 29, 30, 32, 35–39 all