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Vectors in Euclidean Space

A vector in nn-dimensional Euclidean space Rn\mathbb{R}^n is an ordered list of nn real numbers, usually written as a column vector:

v⃗=(v1,v2,…,vn)=[v1v2⋮vn]\vec{v} = (v_1, v_2, \ldots, v_n) = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}

Note: We often use boldface (v\mathbf{v}) or arrows (v⃗\vec{v}) to denote vectors, distinguishing them from scalars (ordinary numbers).

Geometric Interpretation

  • Vectors as points: A vector represent a point in the Euclidean space. In this case, we usually write the vector in the form (v1,…,vn)(v_1,\ldots, v_n).

  • Vectors as arrows: More commonly, we think about a vectors as an arrow. It has both magnitude (length) and direction

Visualizing Vectors in 2D

Drag the arrow head to change direction, the tail to move the start, or the shaft to translate.

u⃗

Vector Arithmetic

Given two vectors u⃗=(u1,u2,…,un)\vec{u} = (u_1, u_2, \ldots, u_n) and v⃗=(v1,v2,…,vn)\vec{v} = (v_1, v_2, \ldots, v_n) in Rn\mathbb{R}^n, and a scalar c∈Rc\in\mathbb{R}. We define

  • Vector addition/subtraction: u⃗±v⃗=(u1±v1,u2±v2,…,un±vn)\vec{u}\pm\vec{v} = (u_1\pm v_1, u_2\pm v_2, \ldots, u_n\pm v_n)
  • Scalar multiplication: cu⃗=(cu1,cu2,…,cun)c\vec{u} = (cu_1, cu_2,\ldots, cu_n)

Think about vectors as displacement.

  • First walk along u⃗\vec{u} from the origin
  • Then walk along v⃗\vec{v} from where you ended
  • You end up at the same place as if you walked along u⃗+v⃗\vec{u} + \vec{v} directly from the origin

Parallelogram Rule for Vector Addition

Drag the arrow head to change direction, the tail to move the start, or the shaft to translate.

u⃗+v⃗u⃗v⃗

Vector subtraction can be understood as adding the opposite vector: u⃗−v⃗=u⃗+(−v⃗)\vec{u} - \vec{v} = \vec{u} + (-\vec{v}).

  1. The red dashed vector −v⃗-\vec{v} is v⃗\vec{v} reversed (opposite direction, same magnitude)
  2. Form a parallelogram with u⃗\vec{u} and −v⃗-\vec{v} as adjacent sides
  3. The purple diagonal from the origin is u⃗−v⃗\vec{u} - \vec{v} (using parallelogram law)

Vector Subtraction

Drag the arrow head to change direction, the tail to move the start, or the shaft to translate.

-v⃗u⃗-v⃗u⃗-v⃗u⃗v⃗

The lighter purple vector from the tip of v⃗\vec{v} to the tip of u⃗\vec{u} also represents u⃗−v⃗\vec{u} - \vec{v}. Think about u⃗\vec{u} as your position and v⃗\vec{v} as a friend's position, then u⃗−v⃗\vec{u} - \vec{v} is the displacement pointing from your friend to you.

Scalar multiplication stretches or shrinks a vector and may reverse its direction:

  • If c>1c > 1: Stretches v⃗\vec{v} by factor cc (same direction)
  • If 0<c<10 < c < 1: Shrinks v⃗\vec{v} by factor cc (same direction)
  • If c=0c = 0: Results in the zero vector 0⃗\vec{0}
  • If c<0c < 0: Reverses direction and scales by ∣c∣|c|

For v⃗=[21]\vec{v} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}:

  • 2v⃗=[42]2\vec{v} = \begin{bmatrix} 4 \\ 2 \end{bmatrix} — twice as long, same direction
  • 12v⃗=[10.5]\frac{1}{2}\vec{v} = \begin{bmatrix} 1 \\ 0.5 \end{bmatrix} — half as long, same direction
  • −v⃗=[−2−1]-\vec{v} = \begin{bmatrix} -2 \\ -1 \end{bmatrix} — same length, opposite direction
  • −3v⃗=[−6−3]-3\vec{v} = \begin{bmatrix} -6 \\ -3 \end{bmatrix} — three times as long, opposite direction

For any vectors u⃗,v⃗,w⃗\vec{u}, \vec{v}, \vec{w} in Rn\mathbb{R}^n:

  1. Commutativity: u⃗+v⃗=v⃗+u⃗\vec{u} + \vec{v} = \vec{v} + \vec{u}
  2. Associativity: (u⃗+v⃗)+w⃗=u⃗+(v⃗+w⃗)(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})
  3. Identity: u⃗+0⃗=u⃗\vec{u} + \vec{0} = \vec{u}
  4. Inverse: u⃗+(−u⃗)=0⃗\vec{u} + (-\vec{u}) = \vec{0}, where −u⃗=[−u1−u2⋮−un]-\vec{u} = \begin{bmatrix} -u_1 \\ -u_2 \\ \vdots \\ -u_n \end{bmatrix}

For any vectors u⃗,v⃗\vec{u}, \vec{v} in Rn\mathbb{R}^n and scalars c,dc, d:

  1. Associativity: c(dv⃗)=(cd)v⃗c(d\vec{v}) = (cd)\vec{v}
  2. Distributivity over vector addition: c(u⃗+v⃗)=cu⃗+cv⃗c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}
  3. Distributivity over scalar addition: (c+d)v⃗=cv⃗+dv⃗(c + d)\vec{v} = c\vec{v} + d\vec{v}
  4. Identity: 1v⃗=v⃗1\vec{v} = \vec{v}

Linear Combinations

A linear combination of vectors v⃗1,v⃗2,…,v⃗k\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_k is an expression of the form:

c1v⃗1+c2v⃗2+⋯+ckv⃗kc_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_k\vec{v}_k

where c1,c2,…,ckc_1, c_2, \ldots, c_k are scalars (called coefficients or weights).

Let v⃗1=[101]\vec{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, v⃗2=[012]\vec{v}_2 = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, and v⃗3=[110]\vec{v}_3 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}.

Compute v⃗1+2v⃗2−v⃗3\vec{v}_1 + 2\vec{v}_2 - \vec{v}_3:

v⃗1+2v⃗2−v⃗3=[101]+2[012]−[110]=[101]+[024]+[−1−10]=[1+0−10+2−11+4+0]=[015]\begin{aligned} \vec{v}_1 + 2\vec{v}_2 - \vec{v}_3 &= \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + 2\begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} - \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \\ &= \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + \begin{bmatrix} 0 \\ 2 \\ 4 \end{bmatrix} + \begin{bmatrix} -1 \\ -1 \\ 0 \end{bmatrix} \\ &= \begin{bmatrix} 1 + 0 - 1 \\ 0 + 2 - 1 \\ 1 + 4 + 0 \end{bmatrix} \\ &= \begin{bmatrix} 0 \\ 1 \\ 5 \end{bmatrix} \end{aligned}