Formula Sheet for Vectors

Vectors are quantities that have both magnitude and direction. They are represented by an arrow, with the length of the arrow representing the magnitude of the vector and the direction of the arrow indicating the direction of the vector. Here are some important formulas related to vectors:

Magnitude of a Vector
The magnitude of a vector is its length. It is denoted by |v|. If the vector is represented by (a, b), then its magnitude is given by:

|v| = √(a² + b²)

Addition of Vectors
When two vectors are added, they are added component-wise. If two vectors are represented by (a, b) and (c, d), then their sum is given by:

(a, b) + (c, d) = (a + c, b + d)

Subtraction of Vectors
When two vectors are subtracted, they are subtracted component-wise. If two vectors are represented by (a, b) and (c, d), then their difference is given by:

(a, b) - (c, d) = (a - c, b - d)

Scalar Multiplication of Vectors
When a vector is multiplied by a scalar, each component of the vector is multiplied by the scalar. If a vector is represented by (a, b) and k is a scalar, then the scalar multiple of the vector is given by:

k(a, b) = (ka, kb)

Dot Product of Vectors
The dot product of two vectors is a scalar quantity. If two vectors are represented by (a, b) and (c, d), then their dot product is given by:

(a, b) · (c, d) = ac + bd

Cross Product of Vectors
The cross product of two vectors is a vector that is perpendicular to both of the original vectors. If two vectors are represented by (a, b, c) and (d, e, f), then their cross product is given by:

(a, b, c) × (d, e, f) = (bf - ce, cd - af, ae - bd)

Unit Vector
A unit vector is a vector that has a magnitude of 1. It is denoted by ̂v. If a vector is represented by (a, b), then its unit vector is given by:

̂v = (a/|v|, b/|v|)

Conclusion
In conclusion, vectors are an important concept in mathematics and physics. These formulas are essential to understanding the properties and behavior of vectors. Understanding these formulas will help in solving problems related to vectors and their applications in various fields.