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Vector Review Workshop
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Introduction to Vectors
A scalar is a quantity that describes magnitude or size only – it does not include direction
A vector is a quantity that has both magnitude and direction
There are 3 ways to represent vectors:
1. Words 5 km at an angle of 30⁰ to the horizontal
2. Diagram
⃑⃑⃑⃑⃑ or 𝐴𝐵
⃑⃑⃑⃑⃑ 𝑣 or 𝑣
3. Symbols 𝐴𝐵
⃑⃑⃑⃑⃑ |
The magnitude of the vector is its length, and it is represented by |𝐴𝐵
Parallel vectors have the same or opposite direction, and may have the same magnitude
Example
Name all the sets of parallel vectors in this shape
A
D
C
B
Equivalent vectors have the same direction and magnitude
Opposite vectors have opposite directions and the same magnitude
Example
Name all the sets of equivalent vectors in this shape. Name all the sets of opposite vectors as well
P
S
R
Q
Vector Direction
True or Azimuth Bearing: A compass measurement where the direction is measured from north in a
clockwise direction, this is always a three digit number without any decimals
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Quadrant Bearing: a measurement between 0⁰ and 90⁰ east or west of the north south line. There
are 3 components of this sort of bearing, and they always go in the following order:
N or S, degree measure, E or W
Again, there are no decimals.
Example
Draw a geometric vector for:
a) 𝑣 = 3 𝑘𝑚 at a bearing of 200⁰
(or S 20⁰ W)
b) 𝑢
⃑ = 2 𝑘𝑚 at a quadrant bearing of S
55⁰ E (or 125⁰)
Adding and Subtracting Vectors
⃑ =𝑅
⃑⃑ )
⃑ +𝑏
Adding Vectors (𝑎
1.
Tip to tail method (or triangle method)
2. Tail to tail method (or parallelogram method)
⃑ ) = ⃑⃑𝑅)
⃑ + (−𝑏
Subtracting Vectors (𝑎
1.
Tip to tail method
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2. Tail to tail method
The zero vector has no direction and a magnitude of zero
Example
Example
Express ⃑⃑⃑⃑⃑
𝐴𝐶 as a sum of two vectors, in as many ways as possible
A
B
E
C
D
Properties of Addition and Subtraction of Vectors
Commutative property: 𝑎 + 𝑏⃑ = 𝑏⃑ + 𝑎
Associative property: (𝑎 + 𝑏⃑) + 𝑐 = 𝑎 + (𝑏⃑ + 𝑐)
⃑ =0
⃑ +𝑎 =𝑎
Identity property:𝑎 + 0
Example
Determine the direction and magnitude of 𝑎 + 𝑏⃑, |𝑎| = 5 𝑘𝑚, |𝑏⃑| = 10 𝑘𝑚
165⁰
𝑏⃑
𝑎
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Example
An airplane travels 300 km due north and then 200 km N 30⁰ E. How far from the starting point did it
travel?
Breaking Down Vectors into Rectangular Components
Example
A frosh pulls a rope attached to a box of textbooks with a force of 200 N. The rope makes an angle of
20⁰ with the ground. Determine the magnitude of the horizontal and vertical components of the
force
Example
A 196 N truck is resting on a ramp inclined at an angle of 15⁰. Resolve the weight into rectangular
vector components that keep the truck at rest.
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Multiplying a Vector by a Scalar
When a vector is multiplied by a scalar, it becomes a scalar multiple of the original vector. The new
vector will be collinear with the original vector. Multiplying a vector by a scalar only changes the
magnitude of the vector, not the direction.
Scalar multiple: 𝑘𝑣 , 𝑘 𝜖 ℝ
Collinear: vectors lying on a straight line when arranged tail to tail
Example
An airplane is flying at a speed of 500 km/h on a heading of 315⁰. The airplane encounters a wind of
120 km/h from a bearing of 065⁰. Determine the ground velocity of the plane and the direction of
flight.
Heading: direction in which a vessel is pointed in order to overcome other forces (e.g. wind) with the
intended resultant direction being the bearing
Ground velocity: velocity of an object relative to the ground
Example
A mass of 196 N is suspended from a ceiling by two lengths of rope that make angles of 60⁰ and 45⁰
with the celling. Determine the tension in the ropes
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Cartesian Vectors
A Cartesian vector is one represented using Cartesian coordinates
⃑⃑⃑⃑⃑
𝑢
⃑ is a vector with endpoints A (-4, -5) and B (-1, -3), so 𝑢
⃑ = 𝐴𝐵
We can translate 𝑢
⃑ so that its tail is at the origin, this then
becomes a position vector.
𝑢
⃑ = [3, 2], this can be used to represent any vector with the
same magnitude and direction, located anywhere on the
Cartesian coordinate system
A unit vector is a vector with a magnitude of 1. 𝑖 = [1, 0,0], 𝑗 = [0,1,0], 𝑘⃑ = [0,0,1] (In 2-space, neglect
the z value, 𝑘⃑ does not exist in 2-space)
𝑘⃑ = [0, 0,1]
𝑖 = [1,0, 0]
𝑗 = [0,1,0]
If we were to resolve 𝑢
⃑ into horizontal and vertical components in terms of 𝑖 and 𝑗, it would be
Adding and Multiplying Cartesian Vectors
𝑢
⃑ = [𝑢1 , 𝑢2 ] and 𝑣 = [𝑣1 , 𝑣2 ]
Adding: 𝑢
⃑ ± 𝑣 = [𝑢1 ± 𝑣1 , 𝑢2 ± 𝑣2 ]
Multiplying: 𝑘𝑢
⃑ = [𝑘𝑢1 , 𝑘𝑢2 ]
Example
⃑ = [1, 2] 𝑣
𝑢
⃑ = [2, −5]
a) 𝑢
⃑ +𝑣
b) 𝑢
⃑ −𝑣
c) 𝑣 − 𝑢
⃑
d) 3𝑢
⃑
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⃑
𝑢
The Unit Vector of 𝑢
⃑ is 𝑢̂ = |𝑢⃑|, The magnitude of 𝑢
⃑ is |𝑢
⃑ | = √𝑢1 2 + 𝑢2 2
Example
Determine the unit vector of 𝑢
⃑ = [1, 2] and 𝑣
⃑ = [2, −5]
Cartesian Vector Between Two Points
𝑅⃑ is the Cartesian vector between two points, 𝑃1 and 𝑃2
⃑⃑⃑⃑⃑⃑⃑ ⃑⃑⃑⃑⃑⃑⃑
⃑⃑⃑⃑⃑⃑⃑⃑
𝑃
1 𝑃2 = 𝑃1 𝑂 + 𝑂𝑃2
𝑃1
⃑⃑⃑⃑⃑⃑⃑1 + ⃑⃑⃑⃑⃑⃑⃑
= −𝑂𝑃
𝑂𝑃2
𝑅⃑
= [−𝑥1 , −𝑦1 ] + [𝑥2 , 𝑦2 ]
𝑃2
= [−𝑥1 + 𝑥2 , −𝑦1 + 𝑦2 ]
= [𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 ]
O
Example
Find the coordinate of 𝑀 if ⃑⃑⃑⃑⃑⃑
𝑃𝑀 = [−6, 5] and 𝑃 = (−2, 3)
Example
A small aircraft is flying at a bearing of 330⁰ at a constant speed of 150 km/h. The wind is blowing at a
bearing of 085⁰ with a speed of 40 km/h. Determine the actual speed and direction of the aircraft
relative to the ground
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Dot Product
You cannot multiply vectors!!!! The outcome of the dot product is a scalar, not a vector! The dot
product can be used in any dimension.
Calculating the Dot Product
The dot product can be calculated in two different ways:
For vectors with bearings:
𝑎 ∙ 𝑏⃑ = |𝑎||𝑏⃑| cos 𝜃
For Cartesian vectors:
𝑎 = [𝑎1 , 𝑎2 ] and 𝑏⃑ = [𝑏1 , 𝑏2 ], 𝑎 ∙ 𝑏⃑ = 𝑎1 𝑏1 + 𝑎2 𝑏2
Properties of the Dot Product





𝑎 and 𝑏⃑ are perpendicular/orthogonal if and only if 𝑎 ∙ 𝑏⃑ = 0
For any vector 𝑎, 𝑎 ∙ 𝑎 = |𝑎|2
For any vectors 𝑎 and 𝑏⃑, 𝑎 ∙ 𝑏⃑ = 𝑏⃑ ∙ 𝑎 (commutative property)
For any vectors 𝑎 and 𝑏⃑, and scalar 𝑘, (𝑘𝑎) ∙ 𝑏⃑ = 𝑘(𝑎 ∙ 𝑏⃑) (associative property)
For any vectors 𝑎, 𝑏⃑, and 𝑣 , 𝑎 ∙ (𝑏⃑ + 𝑣 ) = 𝑎 ∙ 𝑏⃑ + 𝑎 ∙ 𝑣 (distributive property)
Special Cases


If 𝜃=0; 𝑎 ∙ 𝑏⃑ = |𝑎||𝑏⃑|
If 𝜃=180; 𝑎 ∙ 𝑏⃑ = −|𝑎||𝑏⃑|

If 𝜃=90; 𝑎 ∙ 𝑏⃑ = 0
Example
Simplify: a) (𝑎 + 𝑏⃑) ∙ 𝑘𝑎
b) (𝑎 + 𝑏⃑) ∙ (𝑎 − 𝑏⃑)
c) (2𝑢
⃑ + 𝑣 ) ∙ (𝑢
⃑ − 2𝑣 )
Example
Calculate ; 𝑎 ∙ 𝑏⃑ if a = [4,8,10] and b= [9,2,7]
Example
Determine the angle between 𝑎 = [4, 2] and 𝑏⃑ = [−3, 4]
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Example
Another frosh pulls a case of lazy risers 8 m with a force of 245 N at an angle of 18⁰ to the ground.
Find the work the frosh performs
𝑤𝑜𝑟𝑘 = 𝑓 ∙ 𝑠 = |𝑓𝑜𝑟𝑐𝑒 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑖𝑛 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛||𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡| cos 𝜃
Cross Product
The cross product is defined as 𝑎 × 𝑏⃑ = (|𝑎||𝑏⃑| sin 𝜃)𝑛̂, where 𝑛̂ is the unit vector orthogonal to both
𝑎 and 𝑏⃑ (following the right hand rule for direction), and 𝜃 is the angle between the vectors.
The cross product only works in 3 dimensions. They work the exact same way as 2 dimensional
vectors, but with a third coordinate.
There is a 3rd unit vector in 3D, 𝑖̂ = [1, 0, 0], 𝑗̂ = [0, 1, 0], 𝑘̂ = [0, 0, 1]
If 𝑎 = [𝑎1 , 𝑎2 , 𝑎3 ] and 𝑏⃑ = [𝑏1 , 𝑏2 , 𝑏3 ], then,
𝑎 × 𝑏⃑ = [𝑎2 𝑏3 − 𝑎3 𝑏2 , 𝑎3 𝑏1 − 𝑎1 𝑏3 , 𝑎1 𝑏2 − 𝑎2 𝑏1 ]
= (𝑎2 𝑏3 − 𝑎3 𝑏2 )𝑖̂ + (𝑎3 𝑏1 − 𝑎1 𝑏3 )𝑗̂ + (𝑎1 𝑏2 − 𝑎2 𝑏1 )𝑘̂
The magnitude of the cross product, |𝑎 × 𝑏⃑|, is |𝑎 × 𝑏⃑| = |𝑎||𝑏⃑||sin 𝜃|
Trick for Calculating Cross Product
If 𝑎 = [𝑎1 , 𝑎2 , 𝑎3 ] and 𝑏⃑ = [𝑏1 , 𝑏2 , 𝑏3 ]
Each set of crossing arrows represents one term in the resultant. The arrow from the right to left,
downwards is a positive multiplication, and the arrow from left to right, upwards, is a negative
multiplication. These two products are summed to give a term in the resultant vector.
a2b3
-a3b2
a2 a3 a1 a2
b2 b3 b1 b2
term 1 :a2b3 – a3b2
Resultant vector: [𝑎2 𝑏3 − 𝑎3 𝑏2 , 𝑎3 𝑏1 − 𝑎1 𝑏3 , 𝑎1 𝑏2 − 𝑎2 𝑏1 ]
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Properties of the Cross Product
For any vectors 𝑢
⃑ , 𝑣 , and 𝑤
⃑⃑ and scalar 𝑘 𝜖 ℝ




𝑢
⃑ × 𝑣 = −(𝑣 × 𝑢
⃑ ) (The cross product is non-commutative)
𝑢
⃑ × (𝑣 + 𝑤
⃑⃑ ) = 𝑢
⃑ ×𝑣+𝑢
⃑ ×𝑤
⃑⃑
𝑢
⃑ × 𝑣 = 0 if 𝑢
⃑ and 𝑣 are non-zero, and there is a scalar 𝑚 𝜖 ℝ such that 𝑢
⃑ = 𝑚𝑣
𝑘(𝑢
⃑ × 𝑣 ) = 𝑘𝑢
⃑ ×𝑣 =𝑢
⃑ × 𝑘𝑣
Special Cases


𝑢
⃑ ∥ 𝑣 ,𝑢
⃑ ×𝑣 =0
𝑢
⃑ ⊥ 𝑣 ,𝑢
⃑ × 𝑣 = |𝑢
⃑ ||𝑣 |
Tip: You can check whether you have done your cross product properly by calculating the dot
product of the resultant with either of the original vectors. The dot product of the resultant with
both original vectors must equal 0, as the resultant is orthogonal to both vectors.
𝑢
⃑ ×𝑣 = 𝑚
⃑⃑
𝑢
⃑ ∙𝑚
⃑⃑ = 0; 𝑣 ∙ 𝑚
⃑⃑ = 0
Example
Find a vector that is orthogonal 𝑎 = [5, 8, 3] and 𝑏⃑ = [2, 5, 9]
Example
A 10 N force is applied to a wrench, 14 cm from the centre of the bolt. The force is applied in the
counterclockwise direction & makes an angle of 30⁰ with the wrench. What is the magnitude and
direction of the torque?
Torque, 𝜏 is the force acting on an object causing it to rotate, in joules. 𝜏 = 𝑟 × 𝐹 , where 𝑟 is the
distance from the point of rotation, and 𝐹 is the force applied
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1Example
Are the vectors 𝑢
⃑ = [1, 2, −4], 𝑣 = [−5, 3, −7] and 𝑤
⃑⃑ = [−1, 4, 2] are in the same plane?
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Dawkins, P. (2016). Calculus II. Retrieved from Paul's Online Math Notes:
http://tutorial.math.lamar.edu/Problems/CalcII/CrossProduct.aspx
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