9.6 Vectors in Space

9.6 Vectors in Space

Doesn't this --->


look cool?

Vectors 3-D

• Determine distance (not new)

• Position Vector (not new)

• Operations on Vectors (not new)

• Dot Product (not new)

• Angle between vectors (not new)

• Direction Anges

So much is not new...what is?

Vector has 3 directions.....think of the corner of a room or a tissue box.  We now have i, j and k directional vectors.  

check out --->

Distance in space (3D) still uses Pythagorean principles yet we extend it to a third, z-value, as we now have ordered triples for points
(x, y, z).

0(-1, -1, -2)
P(1, 2, 3)

Multiple Choice

Determine the distance.

• $$d=10$$

• $$d=\sqrt{101}$$

• $$d=13$$

• $$d=101$$

Way to go!

yes!

Position Vector

Multiple Choice

Given points A(3, 2, -4) and B(2, 1, -1), determine the position vector of AB

• i + 3j -5k

• -i -3j +3k

• i -j -5k

• -i -j +3k

​

Multiple Choice

$$<1,2>\ +\ <3,4>\ =<4,6>$$  what do you think this is?

$$<1,2,3>+<3,4,5>$$  

• $$<4,6,8>$$  

• $$<-1,\ -1,\ -1>$$  

• $$<0,0,0>$$  

Multiple Choice

Determine u-v

• $$<5,3,4>$$

• $$<11,9,14>$$

• $$<5,-3,\ 4>$$

• $$<0,0,0>$$

Dot Product

• Definition is the same as you know it to be, but it's just extended to one more term

Multiple Choice

If $$\overrightarrow{\ \ a}=3i\ +5j+k\ \ and\ \overrightarrow{\ \ b}=2i+j+3k\ then\ \overrightarrow{\ \ a}\ .\overrightarrow{\ \ b}=$$  

• 41

• 12

• 21

• 14

Unit Vector

Another extension of 2-D vectors.

Multiple Choice

$$v=2i-j+2k$$  

Determine the unit vector of vector v.  

• $$u=\frac{2}{3}i-\frac{1}{3}j+\frac{2}{3}k$$  

• $$u=\frac{2}{\sqrt{5}}i-\frac{1}{\sqrt{5}}j+\frac{2}{\sqrt{5}}k$$  

• $$u=2i-j+2k$$  

Angle Between Vectors

Again Not different just an extension of 2-D vectors....

Multiple Choice

What is the angle between the vectors given?

• 45.3^o

• 111.7^o

• 21.7^o

• 54.6^o

Direction Angles 

Multiple Choice

v=-3i+2j-6k; determine the direction angle for

$$\alpha$$  which means using a=-3 as it takes into account the i direction angle

• $$115.4\degree$$  

• $$64.6\degree$$  

• $$73.4\degree$$  

• $$149.0\degree$$  

Multiple Choice

v=-3i+2j-6k; determine the direction angle for

$$\alpha$$  which means using b=2

• $$115.4\degree$$  

• $$64.6\degree$$  

• $$73.4\degree$$  

• $$149.0\degree$$  

Multiple Choice

v=-3i+2j-6k; determine the direction angle for

$$\alpha$$  

• $$115.4\degree$$  

• $$64.6\degree$$  

• $$73.4\degree$$  

• $$149.0\degree$$  

Go to this website and move the vectors by clicking and dragging them around

https://bit.ly/3cN12L9

All done with Vectors in Space

Just 3D.......how would you summarize this lesson?