Unit 9 Lesson 2

Pre-Calculus Honors

Unit 9 Lesson 2: The Unit Vector and Linear Combinations of Vectors

Objective: ______

1.  Do Now: Represent 3w – 1/2v geometrically. Draw and label each vector.

2. Mark up the following definition for linear combination form of vectors

Definition of Linear Combinations
Component Form: v = < v1, v2
Linear combination Form: v = v1i + v2j
The scalars v1 and v2 are called the horizontal and vertical components of v respectively and can be written as, what is called, a linear combination of vectors i and j.
You can solve vector operation problems by converting u and v from linear combination form to component form. This, however is not necessary. You can perform the rules you learned yesterday to vectors in linear combination form.
For example Let u = -3i +8j in linear combination form is equivalent to u = <-3, 8> in component form.
Name the three ways vectors can be represented:
·  ______
·  ______
·  ______

Group Practice 1: Discovering Component Form Formula, Given a Magnitude and Directional Angle

Diagram / Geometric Representation / Example Problem
Directions: Label the x (the horizontal component), the y (the vertical component), (the magnitude of the vector, and on the diagram below.
/ Directions: Mark up the following text and answer the questions below.
Direction is measured in different ways and in different contexts, especially navigation. A precise way to specify the direction of a vector is the directional angle (the angle that v makes with the positive x-axis.)
1.  Using trigonometry, how can you find the horizontal component of a vector given and ?
x = ______
2.  Using trigonometry, how can you find the vertical component of a vector given and ?
y = ______
3.  Using questions #1 and #2, write a formula that you can use to find the component form of a vector given the directional angle and magnitude.
V = < ______, ______> / Find the component form of a vector v with a directional angle of 115 degrees and a magnitude of 6.
Component Form:
______
Linear Combination Form
______

Group Practice 2: Discovering How to Find the Direction for a Vector

Proof / Example Problem / Geometric Representation
Directions: Now that you know mark up the steps of the proof below.
It follows that the direction angle of for v is determined by
1.)
Step 1: The Definition of Tangent
2.)
Step 2: Multiply the numerator and denominator by the magnitude
3.)
Step 3: Substitute definition in for horizontal component and vertical component
Therefore, you can find the reference angle of the directional angle by .
Step4: Following, you have to calculate the directional angle. / Directions: Find the magnitude and directional angle of the vector
v = -2i -5j / Directions: Draw a diagram that represents the example problem on the left. Label your reference angle, directional angle, and magnitude on the diagram below.

Group Practice 3: Find the vector v with the given magnitude and same direction as u.

Unit 9 Lesson 2 Problem Set

1. Let A = (2, -1), B = (3, 1), C = (-4, 2) and D = (1, -5).

Find the component form and the magnitude of the vector

a) b) c)

2. Use the figure to sketch a graph of the specified vector. Do each example on a separate coordinate plane. Label all vectors.

(a) –3u
/ (b) u + 2 v

(c) 2u -1/2 v
/ (d) 1/4v

3. Find the component form of vector v.

4.  Find the magnitude and directional angle of the vector.

a)  < 3, 4 > b) -3i – 5j c.) 7(cos135◦I + sin135◦j)

5.  Find the vector v with the given magnitude and same direction as u.

llvll = 2, u = < 3, -3 >

Answer Key

#1a < 3 , 6 > Magnitude =

#1b < 3 , -11 > Magnitude =

#1c <-8 , -3 > Magnitude =

#3 <-14.52, 44.70>

#4a Magnitude = 5 53.13 degrees

#4b Magnitude = 239.036 degrees

#4c Magnitude = 7 135 degrees

#5 <1.41, - 1.41>