Vectors

• Vectors are mathematical objects used to define a quantity with both direction and magnitude.
• They are typically denoted by a small letter with an arrow on it (e.g., a⃗ ).
• Vectors can be represented visually as directed line segments. The length of the segment is proportional to the magnitude of the vector and the direction of the arrow shows the direction of the vector.
• The initial point of a vector is called the tail and the end point is called the head.
• The Zero Vector is a vector of zero length and can point in any direction. It is denoted by 0⃗ .
• Vectors are equal if they have the same magnitude and direction, regardless of their initial point. This is known as the Parallelogram Law.
• Vectors can be added and subtracted. Vector addition corresponds to the ‘triangle rule’. Subtraction is the same as adding the negative of the second vector.
• Vectors can be multiplied by a scalar (a real number) to create a new vector. This operation changes the magnitude of the vector but not the direction (unless the scalar is negative).
• In 2D and 3D, vectors can be written in component form: a⃗ = ai⃗ + bj⃗ + ck⃗ , where i⃗, j⃗, and k⃗ are unit vectors in the x, y, and z directions respectively, and a, b, and c are the components along those directions.
• The process of finding the component form of a vector from its initial and terminal point is called ‘Resolving a vector’.
• The distance between the initial and terminal point of a vector in component form in the space can be found using the Pythagorean theorem.

• Trigonometry can be used to find the direction of a vector. The angle θ between two vectors can be found by cosine rule: cos θ =a⃗•b⃗ /

  a⃗

  b⃗

• A unit vector has a magnitude of one. Any vector can be converted into a unit vector by dividing it by its magnitude.
• It’s worth noting that although they’re not primarily involved with Pythagoras and trigonometry, vectors do play an important role in the wider scope of mathematics. Not only do they capture key details about direction and magnitude, but they also help to facilitate complex transformations and provide the groundwork for calculus in higher dimensions.