Analysis
Name_______________________
Vectors
Objectives:
1.
2.
3.
4.
5.
Vector
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Adding Vectors in
the Plane
Graph vectors in a coordinate plane.
Represent vectors in rectangular coordinates.
Perform operations with vectors. (add, subt, scalar mult.)
Find a unit vector that has the same direction as a given non-zero
vector.
Write a vector in terms of its magnitude and direction.
quantity that involves both magnitude (a number indicating size) and
direction.
represented by a directed line segment (arrow)
The magnitude is represented by the length and direction is indicated
by the arrow head.
denoted by a bold letter in print such as u or v.
denoted by an arrow on top of the letter(s) when handwritten
⃗⃗⃗⃗⃗ where P is the initial point (tail) and Q is the
such as u
⃗ ,v
⃗ or 𝑃𝑄
terminal point (head).
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position vectors head to tail
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the resultant vector, u + v, extends from the initial point of u to the
terminal point of v
Sketch:
1. 𝐮 + 𝐯
magnitude
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2. 𝐮 − 𝐯
3. 𝐮 + 𝐯 + 𝐰
also called norm of the vector
denoted by |𝐮| 𝑜𝑟 ‖𝐮‖ is the length of the directed line (the distance
between the initial and terminal points of the vector.
Given 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑖𝑛𝑡 𝑃1 (𝑥1 , 𝑦1 ) 𝑎𝑛𝑑 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑝𝑜𝑖𝑛𝑡 𝑃2 (𝑥2 , 𝑦2 ),
then ‖𝐮‖ = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Find the Magnitude
of AB:
𝐴 = (−2, 3) 𝑎𝑛𝑑 𝐵 = (3, −4)
equal vectors
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vectors that have the same magnitude and direction
i and j unit vectors
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i is the unit vector along the positive x-axis.
j is the unit vector along the positive y-axis.
Position vector
(vector in standard
Position)
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initial point of the vector is the origin
terminal point of the vector is (a, b)
denoted by v = ai + bj or a, b , where a is the horizontal
component and b is the vertical component.
the magnitude of v = ai + bj is given by ‖𝐯‖ = √𝑎2 + 𝑏 2
Sketch each vector as
a position vector and
find its magnitude.
Representing Vectors
In Rectangular
Coordinates
Let v be the vector from
initial point P1 and
terminal point P2 .
Vector v with initial point P1  x1 , y1  and terminal point P2  x2 , y2  is equal
to the position vector v =  x2  x1  i +  y2  y1  j
𝑃1 (5, −1) 𝑎𝑛𝑑 𝑃2 (−2, −3)
v in vector position
v in component form.
(in terms of i and j).
unit vector
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vectors that have a magnitude of 1.
to find a unit vector with same direction as v, divide v by its
𝐯
magnitude ‖𝐯‖ .
Find the unit vector that
has the same direction as
the vector v.
𝐯 = ⟨−3 , 4⟩
𝐮 = ⟨4√3, −2√3 ⟩
Vector Operations
If v = a1 i + b1 j, w = a2 i + b2 j, and k is a real number then
v + w =  a1  a2  i +  b1  b2  j
v - w =  a1  a2  i +  b1  b2  j
kv =  ka  i +  kb  j
Let 𝐯 = 3𝐢 + 5𝐣
𝐰 = −2𝐢 + 𝟑𝐣
find:
𝐯+𝐰
Given 𝐮 = ⟨−5, 7⟩,
𝐯 = ⟨−4, −1⟩
find:
𝐯−𝐮
Direction Angle
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positive angle between the x-axis and a position vector
denoted by 𝜃.
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𝐭𝐚𝐧 𝜽 =
3𝐰
2𝐯 − 𝐰
−4(𝐮 − 𝐯)
𝒃
𝒂
, 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 0
Find the direction angle.
and the magnitude of v.
𝐯 = ⟨−1, 5⟩
Relation of component
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cos 𝜃 = ‖𝐮‖ , 𝑡ℎ𝑢𝑠 𝑎 = ‖𝐮‖ cos 𝜃
to the magnitude
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sin 𝜃 = ‖𝐮‖ , 𝑡ℎ𝑢𝑠 𝑏 = ‖𝐮‖ sin 𝜃
𝑎
𝑏
Writing a vector in
terms of its magnitude
and direction.
Let v be a nonzero vector. If  is the direction angle measured from the
positive x-axis to v, then the vector can be expressed in terms of its
magnitude and direction angle as
𝐯 = ⟨𝑎 , 𝑏⟩ = ⟨‖𝐯‖ cos 𝜃 , ‖𝐯‖ sin 𝜃⟩ or v = ||v|| cos i + ||v|| sin  j
Find the vector given its
direction angle and
magnitude.
‖𝐯‖ = 8 𝑎𝑛𝑑 𝜃 = 330°
‖𝐯‖ = 3 𝑎𝑛𝑑 𝜃 = 85°
Write the vector v in
terms of i and j given
the magnitude and
direction angle.
‖𝐯‖ = 2 𝑎𝑛𝑑 𝜃 = 225°
‖𝐯‖ = 6 𝑎𝑛𝑑 𝜃 = 270°