Vectors

Vector Representation

A vector is a quantity that has both magnitude and direction, often represented as an arrow in a coordinate system or as an ordered pair (x, y) in 2D or (x, y, z) in 3D.

Vector Addition

To add two vectors A and B, position the tail of B at the head of A; the resultant vector C is from the tail of A to the head of B, expressed as C = A + B.

Parallelogram Law

When adding two vectors A and B, the resultant vector can be visualized as the diagonal of a parallelogram formed by the two vectors when placed tail-to-tail.

Triangle Method of Addition

To add vectors A and B using the triangle method, connect the head of A to the tail of B, and the resultant vector C completes the triangle.

Vector Subtraction

Vector subtraction A - B is equivalent to adding the negative of B; this can be expressed as C = A - B = (Ax - Bx, Ay - By).

Scalar Multiplication

When scaling a vector A by a scalar k, if k is positive, the direction remains unchanged but the magnitude is scaled; if k is negative, the magnitude is scaled and the direction is reversed.

Magnitude of a Vector After Scalar Multiplication

The magnitude of the resultant vector from scalar multiplication is calculated as |kA| = |k| * |A|.

Effect of Positive Scalar on Vector

When a positive scalar is applied to a vector, its length is changed proportionally to the scalar while the direction remains the same.

Effect of Negative Scalar on Vector

Applying a negative scalar to a vector alters its magnitude and reverses its direction, resulting in a vector pointing in the opposite direction.