Vectors & Vector Operations Cheat Sheet

Key Facts

• A two-dimensional vector in component form is written as $\vec{v}=\langle a,b\rangle$, where $a$ is the horizontal component and $b$ is the vertical component.
• The magnitude of $\vec{v}=\langle a,b\rangle$ is $\|\vec{v}\|=\sqrt{a^2+b^2}$.
• Vector addition is done component by component: $\langle a,b\rangle+\langle c,d\rangle=\langle a+c,b+d\rangle$.
• Scalar multiplication changes a vector by $k\langle a,b\rangle=\langle ka,kb\rangle$, which changes length by $|k|$ and reverses direction if $k<0$.
• A unit vector in the direction of a nonzero vector $\vec{v}$ is $\hat{v}=\frac{\vec{v}}{\|\vec{v}\|}$.
• The dot product of $\vec{u}=\langle u_1,u_2\rangle$ and $\vec{v}=\langle v_1,v_2\rangle$ is $\vec{u}\cdot\vec{v}=u_1v_1+u_2v_2$.
• The angle between nonzero vectors satisfies $\cos\theta=\frac{\vec{u}\cdot\vec{v}}{\|\vec{u}\|\|\vec{v}\|}$.
• The projection of $\vec{u}$ onto $\vec{v}$ is $\operatorname{proj}_{\vec{v}}\vec{u}=\frac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|^2}\vec{v}$.

Vocabulary

Vector

A quantity with both magnitude and direction, often written as an arrow or in component form such as $\vec{v}=\langle a,b\rangle$.

Magnitude

The length or size of a vector, found in two dimensions by $\|\vec{v}\|=\sqrt{a^2+b^2}$ for $\vec{v}=\langle a,b\rangle$.

Component

One part of a vector along an axis, such as the $x$-component $a$ or the $y$-component $b$ in $\langle a,b\rangle$.

Unit Vector

A vector with magnitude $1$, often found by dividing a nonzero vector by its magnitude: $\hat{v}=\frac{\vec{v}}{\|\vec{v}\|}$.

Dot Product

A scalar result from multiplying corresponding components, given by $\vec{u}\cdot\vec{v}=u_1v_1+u_2v_2$.

Projection

The vector part of one vector that points in the direction of another, calculated by $\operatorname{proj}_{\vec{v}}\vec{u}=\frac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|^2}\vec{v}$.