Vectors Concept
In this course, you will learn how vectors are used to describe movement, direction, and position in mathematics.
First, you will be introduced to vector notation, magnitude, and direction, and learn how to represent vectors using diagrams and coordinates.
Next, you will explore key skills such as vector addition, subtraction, scalar multiplication, and solving geometric problems involving position vectors.
Related Lessons:
Adding Non-Zero Vectors
When vectors form a closed loop but do not all point in the same direction, we can still form a vector equation through addition by following a systematic approach:
- The vectors must form a closed loop, and the vectors are not pointing in the same direction
- Start at a point where two arrows point outwards.
- Follow the first vector in its direction and stop when you reach a point where the next vector is in the opposite direction.
- Then follow the second vector until you arrive at the same endpoint identified in the previous step.
- Equate the two vectors in point (3) and (4)
Adding Resultant Is Zero Vectors
Now, what if all the vectors forming a closed loop point in the same direction? We can still add them by following these steps:
- All the vector arrows point in the same direction.
- We may choose any point as the starting point. For example, let us take an arbitrary point A. Since the vectors form a closed loop, we must also end at point A.
- The overall vector sum is zero since we start and end at the same point and all the vectors are in the same direction.
Free & Position Vectors
What is position vectors?
- Position vectors are vectors that start from the origin, coordinate (0,0)
- If point A has coordinates (2,5), we flip the coordinate point to get the position vector \(\overrightarrow{OA} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\)
- The magnitude of \(\overrightarrow{OA}\) is \(\left| \overrightarrow{OA} \right| = \sqrt{2^2 + 5^2} = \sqrt{29}\)
- Free vectors are any vectors that do not start from the origin and can be express as difference of two position vectors
Parallel Vectors
Two vectors are parallel if their “bracket” values are the same after performing factorisation. For example, vector \(\left(\begin{matrix}6\\14\end{matrix}\right)\) and \(\left(\begin{matrix}3\\7\end{matrix}\right)\) are parallel vectors:
- \(\left(\begin{matrix}6\\14\end{matrix}\right)=2\left(\begin{matrix}3\\7\end{matrix}\right)\) and both vectors have common vector \(\left(\begin{matrix}3\\7\end{matrix}\right)\)
- The second vector has a magnitude 2 times of the first vector