3.13.1 Basic Vectors
3.13.1 Basic Vectors
3.13.1 Basic Vectors
A vector is a mathematical object that represents a quantity that has both magnitude and direction.
A vector can be represented in two ways: geometrically and algebraically.
Geometrically, a vector can be represented as an arrow, where the length of the arrow represents the magnitude of the vector and the direction of the arrow represents the direction of the vector.
Algebraically, a vector can be represented as an ordered list of numbers. E.g.
\( A=(4, 5) , B=(-2,3) \)
Vectors can be added and subtracted, and they can also be multiplied by scalars (numbers).
These operations have geometric interpretations.
For example, adding two vectors geometrically means placing the tail of one vector at the head of the other vector, and the sum vector is the vector that goes from the tail of the first vector to the head of the second vector.
Adding and subtracting vectors:
Adding two vectors is defined geometrically, like this:
Subtraction of one vector from another is equivalent to adding a negative vector.
When vectors are represented as column vectors, adding and subtracting is as simple as adding or subtracting the x and y coordinates of the vectors.
\( a=\binom {2} {4} \), \( b= \binom {3} {2} \)
\( a+b=\binom {2+3} {-4+2}= \binom {5} {-2} \)
\( a-b = \binom {2-3} {-4-2}= \binom {-1} {-6} \)
Magnitude of a vector:
The magnitude of a vector is the length or size of the vector, regardless of its direction.
It is also called the “norm” or “length” of the vector.
The magnitude of a vector is calculated using the Pythagorean theorem.
Given a vector v with components (v1, v2, v3, …, vn), its magnitude ||v|| is:
||v|| = (v12+ v22 + v32 + … + vn2)
\( p=\binom {5} {3} \)
\( |p|=\sqrt{5^2+3^2} \)
\( |p|=5.830951895… \)
Rounding to \( 2 \) decimal places
\( |p|=5.83 \)
Worked example:
\( p=\binom {4} {5} \), \( q=\binom {-2} {7} \)
Find \( 2p+q \)
Multiply both components of \( p \) by \( 2 \)
\( 2p=2\binom {4} {5}=\binom{8} {10} \)
Add together \( 2p \) and \( q \) by adding together corresponding terms
\( 2p+q=\binom {8} {10}+\binom{-2} {7} \)
\( 2p+q=\binom{8-2} {10+7} \)
\( 2p+q=\binom{6} {17} \)
Find \( |p| \)
\( |p| \) means magnitude of \( p \)
The magnitude of a vector can be found using the formula
\( |(\binom {x}{y})|=\sqrt{x^2+y^2} \)
\( |p|=\sqrt{4^2+5^2} \)
\( |p|=6.40312… \)
Round to \( 3 \) significant figures
\( |p|=6.40 \)