Summary of Vectors in the Plane

Vectors are used to represent quantities that have both magnitude and direction.
We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.
Subtraction of vectors is defined in terms of adding the negative of the vector.
A vector is written in component form as ${\bf{v}}=\langle{x,y}\rangle$ .
The magnitude of a vector is a scalar: $\parallel{\bf{v}}\parallel=\sqrt{x^2+y^2}$ .
A unit vector $\bf{u}$ has magnitude $1$ and can be found by dividing a vector by its magnitude: ${\bf{u}}=\frac{1}{\parallel{\bf{v}}\parallel}{\bf{v}}$ The standard unit vectors are ${\bf{i}}=\langle{1,0}\rangle$ and ${\bf{j}}=\langle{0,1}\rangle$ . A vector ${\bf{v}}=\langle{x,y}\rangle$ can be expressed in terms of the standard unit vectors as ${\bf{v}}=x{\bf{i}}+y{\bf{j}}$ .

Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.

Glossary

component: a scalar that describes either the vertical or horizontal direction of a vector

equivalent vectors: vectors that have the same magnitude and the same direction

normalization: using scalar multiplication to find a unit vector with a given direction

parallelogram method: a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram

scalar multiplication: a vector operation that defines the product of a scalar and a vector

standard unit vectors: unit vectors along the coordinate axes: ${\bf{i}}=\langle{1,0}\rangle$ , ${\bf{j}}=\langle{0,1}\rangle$

triangle inequality: the length of any side of a triangle is less than the sum of the lengths of the other two sides

triangle method: a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector

vector difference: the vector difference ${\bf{v}}-{\bf{w}}$ is defined as ${\bf{v}}+(-{\bf{w}})={\bf{v}}+(-1){\bf{w}}$

vector sum: the sum of two vectors, ${\bf{v}}$ and ${\bf{w}}$ can be constructed graphically by placing the initial point of ${\bf{w}}$ at the terminal point of ${\bf{v}}$ ; then the vector sum ${\bf{v}}+{\bf{w}}$ is the vector with an initial point that coincides with the initial point of ${\bf{v}}$ , and with a terminal point that coincides with the terminal point of ${\bf{w}}$