Vectors in R2 and R3

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MCV4U1 – UNIT SIX LESSON FIVE
Lesson Five: Algebraic Vectors in  2 (real numbers, two dimensions) and  3 (real numbers, three dimensions).
In  2 : Consider the vector below with initial point at the origin O(0,0) and terminal point P(a,b).
If iˆ and ˆj are unit vectors in the direction of the positive x- and y-axis, respectively, then we can express vector OP as a
sum, using the triangle law of addition of vectors.

OP  aiˆ  bˆj , where “a” and “b” are called the components of vector OP.
P(a,b)
ĵ
θ
O
iˆ

Using ordered pair notation, we can write OP  a, b  .
So, (a,b) can represent a point or a vector. What the ordered pair represents should be clear to you in the context of the
problem.
When the tail of the vector is at the origin, the vector is called a POSITION VECTOR. The same vector could be translated
ANYWHERE on the grid though, and still be represented by the same ordered pair, (a,b). That is because its magnitude
and direction would still be the same.
Which brings us to……

Magnitude of a vector in algebraic form: OP 
a2  b2
The direction of a vector in algebraic form: The direction of vector OP is given by tanθ, where θ is measured with
respect to the horizontal.
MCV4U1 – UNIT SIX LESSON FIVE
In  3 : Consider the vector below with initial point at the origin O(0,0,0) and terminal point P(a,b,c).
If iˆ , ˆj , and kˆ are unit vectors in the direction of the positive x-, y-, and z-axis, respectively, then we can express vector
OP as a sum, using the triangle law (or head to tail method) of addition of vectors.

OP  aiˆ  bˆj  ckˆ , where “a” , “b”, and “c” are called the components of vector OP.

Using ordered pair notation, we can write OP  a, b, c  .
Note: iˆ, ˆj , and kˆ are often referred to as THE BASIS VECTORS, because all vectors can be written as a linear
combination of these three.

Magnitude of a 3D vector in algebraic form: OP 
a2  b2  c2
The direction of a vector in algebraic form: The direction of 3D vector OP is given by the direction cosines, which is
covered in a later section (p.395 of your book, if you’re interested.)
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