Unit 15 – Day 1 Name: Date: Integrated Math 4G: NOTES Vectors – Introduction to Vectors Geometrically Today, we will begin working with vectors, but I would like to keep everything very informal. No definitions for today, I just want to look at the geometric representations of vectors and work with those. Later in the unit, we will work with more calculations concerning vectors. To the right, there is a picture of a vector.

Vectors will be between two points and one end includes an arrow, while the other end includes no arrow. This picture of a vector reminds you of what other geometric item you have discussed in previous math classes? ________________

Note: For all examples, we will assume 𝑃 is our “starting point” and 𝑄 is our “end point”. Meaning

the “𝑄” end always will get the arrow. Ex 1) Plot the points 𝑃 = (1, 2) and 𝑄 = (5, 5) Ex 2) Plot the points 𝑃 = (6, 4) and 𝑄 = (2, 1) Draw the vector between these points. Draw the vector between these points.

We call this vector 𝑃𝑄⃑⃑⃑⃑ ⃑, which we read as

“vector PQ”.

We call this vector _____, which we read as “________________”.

Find the length of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ Find the length of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ Find the components of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ Find the components of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________

Unit 15 – Day 1

Ex 3) Draw 𝑃𝑄⃑⃑⃑⃑ ⃑ if 𝑃 = (1, 1) and 𝑄 = (4, 3) Draw 𝑅𝑇⃑⃑⃑⃑ ⃑ if 𝑅 = (4, 4) and 𝑇 = (7, 6) Find the length of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ Find the length of 𝑅𝑇⃑⃑⃑⃑ ⃑. __________ Find the components of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ Find the components of 𝑅𝑇⃑⃑⃑⃑ ⃑. __________ What can you say about the two vectors? _________________________________

The location of a vector is rather irrelevant. The points are only used to help draw the vectors. The only two things that really matters for a vector are its length and where the arrow points.

If those two items match (represented by the components), then the two vectors are equal. Equal vectors are also ________________ to each other.

Vectors can also be named using lowercase letters. Ex 4) Draw the following vectors where �⃑⃑�

begins on the point 𝑣 ends: 𝑣 = ⟨3, 6⟩ and �⃑⃑� = ⟨4, 8⟩.

Ex 5) Draw the following vectors where �⃑⃑� begins on the point 𝑣 ends: 𝑣 = ⟨2, 7⟩ and �⃑⃑� = ⟨5, −3⟩.

To add the vectors, complete the triangle and find the components. To add the vectors, complete the triangle

and find the components.

Unit 15 – Day 1 Name: Date: Integrated Math 4G: HMWK

Vectors – Introduction to Vectors Geometrically (#1-3) Given points 𝑃 and 𝑄, draw the vector and find the length and components. 1) Plot the points 𝑃 = (5, 3) and 𝑄 = (1, 2) 2) Plot the points 𝑃 = (−1, 5) and 𝑄 = (6, 1) Draw the vector between these points. Draw the vector between these points.

Find the length of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ Find the length of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________

Find the components of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ Find the components of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ 3) Plot the points 𝑃 = (−3,−4) and 𝑄 = (9, 10)

Draw the vector between these points. Find the length of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________ Find the components of 𝑃𝑄⃑⃑⃑⃑ ⃑. __________

Unit 15 – Day 1 (#4) Answer the questions based on the given information. 4) Based on the figure to the right, what can be said about each set of vectors? a) 𝑧 and 𝑦

b) 𝑥 and �⃑⃑� c) �⃑� and 𝑣 d) 𝑦 and 𝑣 (#5-6) Add the following vectors graphically. 5) Draw the following vectors where �⃑⃑� begins

on the point 𝑣 ends: 𝑣 = ⟨8, 2⟩ and �⃑⃑� = ⟨4, 5⟩.

6) Draw the following vectors where �⃑⃑� begins on the point 𝑣 ends: 𝑣 = ⟨−6,−3⟩ and �⃑⃑� = ⟨3, 8⟩.

Add the vectors graphically. Find the components of 𝑣 + �⃑⃑� . _________

Add the vectors graphically. Find the components of 𝑣 + �⃑⃑� . _________