Vector Operations

A vector is a mathematical object that has both a magnitude (length) and a direction. It is often represented as an ordered pair (x, y) in a two-dimensional space.

Vector addition is the process of adding two or more vectors together to get a resultant vector. This is done by adding their corresponding components.

v + w = <3 + (-2), -5 + 3> = <1, -2>.

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction.

3u = 3 * <5, 2> = <15, 6>.

The zero vector is a vector with a magnitude of zero and no specific direction, represented as <0, 0> in two-dimensional space.

The resultant vector R of two vectors A and B is given by R = A + B, where R = .

2w + y = 2 * <2, -5> + <2, 0> = <4, -10> + <2, 0> = <6, -10>.

Two vectors are equal if they have the same magnitude and direction, meaning their corresponding components are equal.

3y - 2z = 3 * <2, 0> - 2 * <-1, -4> = <6, 0> - <2, 8> = <4, -8>.