GeometryLesson 8 – 7

Vectors

Objective:Find magnitudes and directions of vectors.

Add and subtract vectors.

VectorA quantity that has both magnitude and direction.What is magnitude?The length of the vector from its intial point

to its terminal point.

What is direction?The angle that is formed with the x-axis or

any other horizontal line.

Vector

Standard positionA vector is standard position when its initial point is at the origin.

Component FormDescribed in terms of its horizontal change x and vertical change y from initial point to terminal point.

Component form:

Write the component form of CD

2,4

What is the change in x?What is the change in y?

Write the component form of vector PQ.

4,2

3,4

Find the magnitude and direction of vector RT for R (2, 4) and T(-3, -2)

Magnitude same as length same as distance.How do we find distance?

22 4223

8.761

Cont…

Find direction:Graph vector.

56tan R

56tan 1R

2.50RCont…

Direction is the angle the vector makes with the positive x axis.

Reposition the vector so that it is on the origin.

So vector RT has a Magnitude of 7.8 units

And a direction of about 230.2 degrees.

Find the magnitude and direction.

Cont…

22 7243 d

6.8

Direction

57tan S

S

57tan 1

5.54S270 + 54.5 = 324.5

Magnitude is 8.6 units, direction about 324.5 degrees.

Vector AdditionEqual vectors Two vectors are equal if and only if they have the same

magnitude and direction.

Parallel vectors Two vectors are parallel if and only if they have the same or opposite direction.

Opposite vectors Two vectors are opposites if they have the same magnitude

and opposite directions.

Vector Addition:Parallelogram Method

Step 1Place both vectors at the same initial point.

Step 2Complete the parallelogram. The resultant is the diagonal of the

parallelogram.

Vector Addition:Triangle Method

Step 1Place the initial point of the second vector

at the terminal point of the first.

Step 2The resultant connects the initial point of

the first vector and the terminal point of the second.

Vector Addition and Subtraction.

Find ba

ab

ba

Vector Addition and Subtraction.

ba

Subtracting a vector is like Adding its opposite.

ab

ba

Find

Vector Addition and Subtraction.

ab Find

ab

ab

Vectors can be added algebraically.

Cheerleaders are using an air launcher to project T-shirts into the stands. The wind is blowing in the same direction that the shirts are launched. If a shirt leaves the launcher at 40 feet per second at an angle of 45o and the wind is blowing 7 feet per second parallel to the horizontal, what is the resultant velocity and direction of the T-shirt?

Cont…

4045cos x

3.28x40

45sin y

3.28x3.28,3.28 Add 7 for wind

3.28,3.35