NOTE: “Vectors do not have location”.
Vectors are simply change. They are bind to where it starts. For eg: the vector that represents change of Angad’s
weight change from (2010,30) to (2022,71) will be identical to the vector from Question 1.
Hence, as long as vectors have the same size and direction, they are considered to be same vector.
These are all same vectors:
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Vector VS Scalar
* Vectors have size magnitude and direction, whereas scalar do not have (Direction)
* Vectors are pretty much used to represent change, whereas scalars are mainly used to represent (state) and (size).
Question 2. Identify whether the following quantity are vector or scalar
Mass:
_______
Change in Mass:
Height:
_______
Change in Height:
Velocity:
Speed:
_______
_______
_______
_______
Operation of Vectors
Addition of Vectors
* We cannot add vectors with scalars. Only vectors with same dimension can be added.
* When two vectors are added, the sum vector represents Overall change.
* Algebraically, we collect the like terms for each component of vector (finding the overall change for each
component of vectors).
From Zain’s booklet:
*The overall change of vectors 𝑚 and 𝑛 are increasing 𝑎 component by 9 increasing 𝑏 component by 1.
* Visually, we line up the vectors and find the overall vector by joining the very start to very end.
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If you could make a better version of that drawing, that would be great. Thank you (if not, its’s all good).
Subtraction of Vectors
*For the subtraction of vectors i.e 𝑢 − 𝑣, we can instead add vectors 𝑢 and −𝑣 instead.
Scalar Multiplications
* A scalar and a vector can be multiplied together.
* It results in stretching or shrinking thevector.
* If the scalar is negative, it causes the vector to face opposite way.
* Algebraically
𝑚 = 4𝑎 − 3𝑏
2𝑚 = 8𝑎 − 6𝑏
*Visually, vector 𝑚 looks like this:
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Question 4. Have a go at drawing the following 
2𝑚
1
𝑚
2
−𝑚
−2𝑚
*NOTE BOX:
When performing scalar multiplication to the vector, the result is always parallel to the original vector.
Algebraically, this is because we are NOT changing the ratio of components as we are multiplying same number to all
components.
The ratio of components is what makes up direction of vectors.
Position Vector
*A position vector is simply a vector that describes the position of a point.
It does so by giving the position change required to get to a point from the origin.
*I.E Position vector of point P (2,4) is ⃗⃗⃗⃗⃗
𝑂𝑃 = 2𝑖 + 4𝑗
Question . Consider a vector which takes us to the position of a point (𝑥, 𝑦) from the origin; position vector of (𝑥, 𝑦).
What would be the position vector of the transformed point if the point has been transformed by:

Dilation of factor 2 from the 𝑥 axis,
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Dilation of factor 3 from 𝑦 axis,
Translated 2 in the positive direction of 𝑥 axis,
 and translated 3 in the position direction of 𝑦 axis?
Space for personal notes
Section B: Vector Geometry
Linear Dependence and Independence
Definition:
●
A set of vectors satisfying the following equation is called Linearly Dependent:
[𝑘1𝑉1 + 𝑘2𝑉2 … . +𝑘𝑛𝑉𝑛 = 0], 𝑤ℎ𝑒𝑟𝑒 𝑘𝑖 ≠ 0
Can you put a box around the formula above (so it stands out to students)
⇨ Otherwise, the set of vectors called Linearly Independent.
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Tutor: James Lee
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Understanding:
●
A set (collection) of vectors is Linearly Dependent if the vectors align in one linear plane
or if the vectors’ directions can close up a linear plane.
Let’s consider the following 3 vectors 𝑎, 𝑏, 𝑐.
Let’s see if the direction of 𝑎, 𝑏, 𝑐 can close up a linear plane by stretching their lengths
And we can see they do
Hencethe set of vectors 𝑎, 𝑏, 𝑐 is linearly dependent.
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