12.3 Vectors in 3d

Definition

• A vector is a quantity with both a magnitude and a direction

  • not to be confused with scalar that is only a magnitude

Representations of a vector

• A vector $\vec{v}$ can be denoted by an arrow in ${\mathbb{R}}^n$, but we must denote its length as $|\vec{v}|$ (to show its direction and magnitude)

  • vectors can be represented any where as long (we usually use the origin because its easy)

  • In ${\mathbb{R}}^2$ its called an ordered pair

  • in ${\mathbb{R}}^3$ its called a ordered triple

• vectors in ${\mathbb{R}}^2,{\mathbb{R}}^3$

  • $\vec{v}=(a,b),{\mathbb{R}}^2$

  • $\vec{v}=(a,b,c),{\mathbb{R}}^3$

    • these numbers a b c are called components of the vector
    • the length of $\vec{v}$ can be found as just using Pythagoras

• position vector:

  • The vector $\vec{v}$ is called the position vector of the point

    • $P(a,b),\text{ or }P(a,b,c)$

Special vectors

• The $\vec{0}$

  • it is special because it has no direction

• Unit vector

  • geometrically the unit vector is just the vector of 1 in any direction (this encodes the information of the vector)

• standard basis vectors

  • $\hat{i}=(0,0,1),\hat{j}=(0,1,0),\hat{k}=(1,0,0)$
    • same ones in physics

Algebraic operations with vectors:

• Addition:

  • $\vec{u}+\vec{v}=({\vec{u}}_1+{\vec{v}}_1)+({\vec{u}}_2+{\vec{v}}_2)+({\vec{u}}_3+{\vec{v}}_3)\dots ({\vec{u}}_n+{\vec{v}}_n)$
  • geometrically its just the A→C line
    • 

• Scalar multiplication

  • if c is a real number you can just extrude the vector

• Difference

  • $\vec{u}-\vec{v}=\vec{u}+(-\vec{v})$
  • geometrically its similar to addition but we put the 2 initial positions together instead of stacking them

Examples

How to find all possible values of a variable to be a unit vector

• find all possible values of $\alpha$ to make $\vec{u}=\left(\frac{1}{4},-\frac{1}{2},\alpha \right)$

$$\begin{array}{r}|\vec{u}{|}^2=1\\ {\left(\frac{1}{4}\right)}^2+\left(-\frac{1}{2}\right)+{\alpha }^2=1\end{array}$$