Welcome to our exploration of vectors! A vector is a fundamental mathematical object that has both magnitude and direction. Unlike scalars such as temperature or mass which only have size, vectors represent quantities like velocity, force, and displacement where direction matters just as much as magnitude.
Vectors can be represented in two main ways. Geometrically, a vector is shown as an arrow where the length represents the magnitude and the arrow points in the direction. Algebraically, we use components. In two dimensions, vector v equals angle bracket 3 comma 2 angle bracket, where 3 is the x-component and 2 is the y-component, showing how far the vector extends along each axis.
Vector addition follows a simple geometric rule called the triangle rule. To add vector a and vector b, we place the tail of vector b at the head of vector a. The sum is the vector from the tail of a to the head of b. Algebraically, we add corresponding components: angle bracket 2 comma 1 angle bracket plus angle bracket 1 comma 2 angle bracket equals angle bracket 3 comma 3 angle bracket.
The magnitude of a vector is its length, calculated using the Pythagorean theorem: magnitude of v equals square root of x squared plus y squared. Scalar multiplication changes the vector's size and possibly direction. Multiplying by 2 doubles the length, multiplying by 0.5 halves it, and multiplying by negative 1 reverses the direction while keeping the same magnitude.
To summarize what we have learned about vectors: Vectors are mathematical objects with both magnitude and direction. They can be represented geometrically as arrows or algebraically using components. Vector addition follows the triangle rule, and scalar multiplication changes their size and direction. Vectors are fundamental tools used extensively in physics, engineering, and computer graphics.