A 2D vector is a mathematical object that has both magnitude and direction in a two-dimensional plane. It can be visualized as an arrow pointing from one point to another. The vector shown here has components x equals 3 and y equals 2, representing movement 3 units right and 2 units up from the origin.
The magnitude of a vector is its length, calculated using the Pythagorean theorem as the square root of x squared plus y squared. For our vector with components 3 and 2, the magnitude is square root of 13, approximately 3.6 units. The direction is the angle the vector makes with the positive x-axis, calculated using arctangent of y over x, which gives us approximately 33.7 degrees.
Vector addition is performed by adding corresponding components. To add two vectors u and v, we add their x-components and y-components separately. Geometrically, we can visualize this by placing the tail of the second vector at the head of the first vector. The sum is the vector from the origin to the final point. In our example, vector u equals 2, 1 and vector v equals 3, 2, so their sum is 5, 3.
Scalar multiplication involves multiplying a vector by a real number called a scalar. This operation multiplies each component of the vector by the scalar. When the scalar is greater than 1, it stretches the vector. When the scalar is between 0 and 1, it shrinks the vector. When the scalar is negative, it reverses the direction. For example, multiplying vector 3, 2 by scalar 2 gives us vector 6, 4.
To summarize what we have learned about 2D vectors: They are mathematical objects with both magnitude and direction in a two-dimensional plane. Vectors can be represented by their x and y components or visualized as arrows. The magnitude is calculated using the Pythagorean theorem, addition combines corresponding components, and scalar multiplication scales both magnitude and direction.