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Wize University Linear Algebra Textbook > Vectors

Vector Properties

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Vector Properties

You may be asked to prove some of the following properties, or to use these properties to prove other statements.

Let u⃗, v⃗, w⃗ ∈Rn\vec{u},\ \vec{v},\ \vec{w}\ \in \reals^n be three vectors.
Let c, d∈Rc,\ d \in \reals be two scalars.
  • u⃗+v⃗=v⃗+u⃗\vec{u}+\vec{v}=\vec{v}+\vec{u}
  • (u⃗+v⃗)+w⃗=u⃗+(v⃗+w⃗)\left(\vec{u}+\vec{v}\right)+\vec{w}=\vec{u}+\left(\vec{v}+\vec{w}\right)
  • There exists a zero vector 0⃗=⟨0,0,...,0⟩∈Rn\vec{0} = \lang 0,0,...,0 \rang \in \reals^n such that v⃗+0⃗=v⃗\vec{v}+\vec{0}=\vec{v}
  • There exists a negative vector −v⃗-\vec{v} such that v⃗+(−v⃗)=0⃗\vec{v}+\left(-\vec{v}\right)=\vec{0}
  • (cd)v⃗=c(dv⃗)\left(cd\right)\vec{v}=c\left(d\vec{v}\right)
  • (c+d)v⃗=cv⃗+dv⃗\left(c+d\right)\vec{v}=c\vec{v}+d\vec{v}
  • c(v⃗+u⃗)=cv⃗+cu⃗c\left(\vec{v}+\vec{u}\right)=c\vec{v}+c\vec{u}
  • 1v⃗=v⃗1\vec{v}=\vec{v}
  • 0v⃗=0⃗0\vec{v}=\vec{0}
Wize Tip
  • To prove a statement involving vectors, it is often helpful to write out the components of each vector.
u⃗=[u1u2⋮un],v⃗=[v1v2⋮vn]\vec u = \begin{bmatrix} u_1\\ u_2\\ \vdots\\ u_n \end{bmatrix}, \quad \vec v = \begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{bmatrix}

  • To show that a statement is NOT always true, find one counter example (keep it simple with an example from R2\reals^2)

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Example: Proving Vector Properties

Given vectors u⃗, v⃗∈Rn\vec u, \ \vec v \in \reals^n and scalar c∈Rc \in \reals, prove the following property:
c(u⃗+v⃗)=cu⃗+cv⃗c(\vec u + \vec v) = c \vec u + c \vec v

Begin by writing out the components of u⃗\vec u and v⃗\vec{v}:
u⃗=[u1u2⋮un],v⃗=[v1v2⋮vn]\vec u = \begin{bmatrix} u_1\\ u_2\\ \vdots\\ u_n \end{bmatrix}, \quad \vec v = \begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{bmatrix}
Then:
c(u⃗+v⃗)=c([u1u2⋮un]+[v1v2⋮vn])=c[u1+v1u2+v2⋮un+vn]=   [c(u1+v1)c(u2+v2)⋮c(un+vn)]=   [cu1+cv1cu2+cv2⋮cun+cvn]=c[u1u2⋮un]+c[v1v2⋮vn]=cu⃗+cv⃗\begin{aligned} c(\vec u + \vec v) = c & \left( \begin{bmatrix} u_1\\ u_2\\ \vdots\\ u_n \end{bmatrix} + \begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{bmatrix} \right)\\ = c &\begin{bmatrix} u_1 +v_1\\ u_2+v_2\\ \vdots\\ u_n+v_n \end{bmatrix}\\ =\ \ \ &\begin{bmatrix} c(u_1 +v_1)\\ c(u_2+v_2)\\ \vdots\\ c(u_n+v_n) \end{bmatrix}\\ =\ \ \ &\begin{bmatrix} cu_1 +cv_1\\ cu_2+cv_2\\ \vdots\\ cu_n+cv_n \end{bmatrix}\\ = c &\begin{bmatrix} u_1\\ u_2\\ \vdots\\ u_n \end{bmatrix} + c \begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{bmatrix}\\ = c &\vec u + c \vec v \end{aligned}
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Example: Proving Vector Properties

Determine whether the following statement is always true.
If it is true, prove it. If it is not always true, provide a counterexample.

For any vectors u⃗, v⃗, w⃗∈Rn\vec u, \ \vec v, \ \vec w \in \reals^n,
u⃗−(v⃗−w⃗)=(u⃗−v⃗)−w⃗\vec u - (\vec v - \vec w) = (\vec u - \vec v) - \vec w

Not always true.

Intuitively, we might think something isn't quite right when we try to distribute the negative on the LHS:
u⃗−(v⃗−w⃗)=u⃗−v⃗+w⃗ (???)\vec u - (\vec v - \vec w) = \vec u - \vec v + \vec w \qquad \text{ (???)}
This leads us to think of a counterexample. Keep it simple with vectors in R2\reals^2.

Let u⃗=[11],v⃗=[22],w⃗=[33]\vec u = \begin{bmatrix} 1\\1 \end{bmatrix}, \quad \vec v = \begin{bmatrix} 2\\2 \end{bmatrix}, \quad \vec w = \begin{bmatrix} 3\\3 \end{bmatrix}.

LHS
u⃗−(v⃗−w⃗)=[11]−([22]−[33])=[11]−[−1−1]=[22]\vec u - (\vec v - \vec w) = \begin{bmatrix} 1\\1 \end{bmatrix} - \left( \begin{bmatrix} 2\\2 \end{bmatrix} - \begin{bmatrix} 3\\3 \end{bmatrix} \right) = \begin{bmatrix} 1\\1 \end{bmatrix} - \begin{bmatrix} -1\\-1 \end{bmatrix} = \begin{bmatrix} 2\\2 \end{bmatrix}

RHS
(u⃗−v⃗)−w⃗=([11]−[22])−[33]=[−1−1]−[33]=[−4−4](\vec u - \vec v) - \vec w = \left( \begin{bmatrix} 1\\1 \end{bmatrix} - \begin{bmatrix} 2\\2 \end{bmatrix} \right) - \begin{bmatrix} 3\\3 \end{bmatrix} = \begin{bmatrix} -1\\-1 \end{bmatrix} - \begin{bmatrix} 3\\3 \end{bmatrix} = \begin{bmatrix} -4\\-4 \end{bmatrix}

Since LHS≠RHS\text{LHS} \neq \text{RHS}, this counterexample implies that the statement is not true in general.

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Practice: Vector Properties

Prove the following vector property:
u⃗+v⃗=v⃗+u⃗\vec{u}+\vec{v}=\vec{v}+\vec{u}

Extra Practice
Practice: Vector Addition and Subtraction
The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 45
Given the points P=(3, 4, −1)\bcb{P = (3,\, 4,\, -1)} and Q=(−1, 2, 2)\bcb{Q = (-1,\, 2,\, 2)}, and the vector u⃗=PQ⃗\bcb{\ol{u} = \vec{PQ}}, find a vector parallel but in the opposite direction to u⃗\bcb{\vec{u}}.
Length / magnitude / norm. 19.1W
1. Find the vector d⃗\vd connecting the points P1(−2,−1)P_1(-2,-1) and P2(3,11)P_2(3,11).
2. Find the length of the vector d⃗\vd found, above. What does this length represent?
A parallelogram in R3R^3 has vertices A(−1,1,2)A\left(-1,1,2\right), B(1,−2,8)B\left(1,-2,8\right) and C=(3,−1,2)C=\left(3,-1,2\right).
Vector Arithmetic
Suppose X=(x1,x2,x3)X=(x_1, x_2, x_3) and Y=(y1,y2,y3)Y=(y_1, y_2, y_3) If −XY⃗=<43,−35,103>\vec{-XY}=\left<\frac{4}{3},\frac{-3}{5}, \frac{10}{3}\right>, what is YX⃗\vec{YX}?
Practice Question 9: Vector Operations
Practice Question: Vector Operations
Let u⃗=(−9, 6)\vec{u}=\left(-9,\ 6\right) and v⃗=(1,−1)\vec{v}=\left(1,-1\right). If we start at the origin and travel along a directed line segment that is opposite to u⃗\vec{u} but has 1/3 of its length, then we travel along the translation of v⃗\vec{v} to your current position. Finally, we travel along the translation of the unit vector that points in the same direction as v⃗\vec{v}.
a.) Where do we end up?
If α∈R\alpha \in \mathbb{R} and v∈Vv \in V , where VV is a real vector space and αv=0\alpha v = 0 , then prove that either α=0\alpha = 0 or v=0v = 0 .
Practice: Vector Addition and Subtraction
The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.
Practice: Magnitude of a Vector
Practice: Magnitude of a Vector
An aircraft's location is 200 miles from base, at a direction of 30° W of S, at an altitude of 35,000 ft. Find the distance between the aircraft and the base.
Note: 1 mile = 5280 feet
Practice Question: Vector Properties
Practice Question: Vector Properties
Given that u⃗\vec{u} and v⃗\vec{v} are vectors in R3\mathbb{R}^3, which of the following statements is/are always true?
Practice: Vector Operations
A parallelogram has sides ABAB, BCBC, CDCD and DADA
The following coordinates are known: A(2,0,1)A(2,0,1), C(6,0,0)C(6,0,0), and M(3,1,0)M(3,1,0), where MM is the midpoint between AA and BB
Find the vector BD⃗\vec{BD}