Vector Operations

Vector Operations and Linear Combinations

7 Activities

Practice: Collinear Vectors
Find the value of ccc such that the vectors u⃗=⟨−3,c,3⟩\vec u = \lang −3,c,3 \rangu=⟨−3,c,3⟩ and v⃗=⟨1,−2,−1⟩\vec v = \lang 1,−2,−1 \rangv=⟨1,−2,−1⟩ are collinear.

c=

I don't know

Practice Question

Is the vector (π,e,1)(\pi, e, 1)(π,e,1) in the linear span of  (1,2,3),(−1,0,1)(1, 2, 3), (-1, 0, 1)(1,2,3),(−1,0,1) and  (5,1,−2)(5, 1, -2)(5,1,−2) of  R3\mathbb{R}^3R3 ? Demonstrate a linear combination that does this.

Find the value of h so that (1,-6,4,h) is a linear combination of (1,-2,1,-4), (0,-4,-1,3) and (0,0,2,5).

Linearly independent vectors

Is this set of vectors {[1 3], [2,-2]} linearly independent?   Write [4,-2] as  a linear combination of this set.

Practice: Linear Combination

Show that if u⃗∈Rn\vec u \in \mathbb{R}^nu∈Rn is a linear combination of x⃗,y⃗∈Rn\vec x,\vec y \in \mathbb{R}^nx,y​∈Rn and v⃗∈Rn\vec v \in \mathbb{R}^nv∈Rn is a (different) linear combination of the same  x⃗,y⃗\vec x,\vec yx,y​ then every linear combination  au⃗+bv⃗a\vec u+b\vec vau+bv  for a,b∈Ra,b\in\mathbb{R}a,b∈R is also a linear combination of x⃗,y⃗\vec x,\vec yx,y​ .

133 - FML 3 - 18.1W e.g. 23

Give a mathematical expression that mean v⃗\bcb{\vec{v}}v and w⃗\bcb{\vec{w}}w are parallel.

Practice: Vector Operations

Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right]u=[1, 0, −1], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right]v=[−2, 4, 1], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right]w=[3, 9, −12], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}5u−21​v+32​w.

Practice: Vector Operations

Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right]u=[1, 0, −1], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right]v=[−2, 4, 1], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right]w=[3, 9, −12], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}5u−21​v+32​w.

Linear combinations

If u⃗=[5−23]\vec{u}=\begin{bmatrix}5\\-2\\3\end{bmatrix}u=​5−23​​is a linear combination of  v⃗=[−2k3]\vec{v}=\begin{bmatrix}-2\\k\\3\end{bmatrix}v=​−2k3​​and w⃗=[32−1]\vec{w}=\begin{bmatrix}3\\2\\-1\end{bmatrix}w=​32−1​​, what is the value of  kkk?

Write 1 Vector as linear combination of others

Write (16,−15,8)(16,-15, 8)(16,−15,8) as a linear combination of  (1,1,1),(1,1,0),(1,0,−1)(1,1,1), (1,1,0),(1,0,-1)(1,1,1),(1,1,0),(1,0,−1), or show that it is not possible to do so.

133 - FML 3 - 18.1W e.g. 49

 Given v⃗=<0, 4, 3>\bcb{\ol{v} = \left< 0,\, 4,\, 3 \right>}v=⟨0,4,3⟩, find a unit vector perpendicular to v⃗\bcb{\vec{v}}v.

133 - FML 3 - 18.1W e.g. 42

 Find a vector parallel to u⃗=<1, −1, 1>\bcb{\vec{u} = \left< 1,\, -1,\, 1\right>}u=⟨1,−1,1⟩. 

133 - FML 3 - 18.1W e.g. 45

Given the points P=(3, 4, −1)\bcb{P = (3,\, 4,\, -1)}P=(3,4,−1) and Q=(−1, 2, 2)\bcb{Q = (-1,\, 2,\, 2)}Q=(−1,2,2), and the vector u⃗=PQ⃗\bcb{\ol{u} = \vec{PQ}}u=PQ​, find a vector parallel but in the opposite direction to u⃗\bcb{\vec{u}}u.

19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{6}$_$\key{|Strc>5.2}$

Describe the set of all linear combinations (sums of constant multiples of) the vectors:
   v⃗=[111]\vv = \colvecth{1}{1}{1}v=​111​​  and  w⃗=[−121]\vw = \colvecth{-1}{2}{1}w=​−121​​. 

Practice Question: When is 1 Vector a Linear Combination of Others (Has a Solution)

Write (6,5,1)(6,5,1)(6,5,1) as a linear combination of (−2,−1,1),(2,9,15),(2,3,3)(-2,-1,1), (2,9,15), (2,3,3)(−2,−1,1),(2,9,15),(2,3,3) , or show that it is not possible to do so.

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 49

 Given v⃗=<0, 4, 3>\bcb{\ol{v} = \left< 0,\, 4,\, 3 \right>}v=⟨0,4,3⟩, find a unit vector perpendicular to v⃗\bcb{\vec{v}}v.

Practice Question: When is 1 Vector a Linear Combination of Others

Is it possible to write down u⃗=(1,2,3)\vec{u}=(1,2,3) u=(1,2,3)as a linear combination of v1⃗=(1,0,2),v2⃗=(0,1,2) and v3⃗=(2,1,6)\vec{v_1}=(1,0,2),\vec{v_2}=(0,1,2)\text{ and }\vec{v_3}=(2,1,6)v1​​=(1,0,2),v2​​=(0,1,2) and v3​​=(2,1,6)? If yes write down u⃗\vec{u}uas a linear combination of other vectors. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 45

Given the points P=(3, 4, −1)\bcb{P = (3,\, 4,\, -1)}P=(3,4,−1) and Q=(−1, 2, 2)\bcb{Q = (-1,\, 2,\, 2)}Q=(−1,2,2), and the vector u⃗=PQ⃗\bcb{\ol{u} = \vec{PQ}}u=PQ​, find a vector parallel but in the opposite direction to u⃗\bcb{\vec{u}}u.

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 42

 Find a vector parallel to u⃗=<1, −1, 1>\bcb{\vec{u} = \left< 1,\, -1,\, 1\right>}u=⟨1,−1,1⟩. 

Find the value of h so that (1,-6,4,h) is a linear combination of (1,-2,1,-4), (0,-4,-1,3) and (0,0,2,5).

19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{4}$_$\key{|Strc>5.2}$

Describe the set of all constant multiples of the vector:
  v⃗=[−121]\vv = \colvecth{-1}{2}{1}v=​−121​​. 

19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{2}$_$\key{|Strc>5.2}$

Describe the set of all constant multiples of the vector v⃗=[11]\vv = \colvec{1}{1}v=[11​]. 

Practice: Properties of Vector Operations

Practice: Properties of Vector Operations
If u⃗, v⃗, w⃗ ∈R3\vec{u},\ \vec{v},\ \vec{w}\ \in R^3u, v, w ∈R3, which of the following statements is/are always true?
i) ∣∣v⃗−u⃗∣∣=−∣∣u⃗−v⃗∣∣\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|∣∣v−u∣∣=−∣∣u−v∣∣

Vector Addition and Scalar Multiplication

Vector Operations

Practice: Collinear Vectors
Find the value of ccc such that the vectors u⃗=⟨−3,c,3⟩\vec u = \lang −3,c,3 \rangu=⟨−3,c,3⟩ and v⃗=⟨1,−2,−1⟩\vec v = \lang 1,−2,−1 \rangv=⟨1,−2,−1⟩ are collinear.

Practice: Vector Operations

Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right]u=[1, 0, −1], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right]v=[−2, 4, 1], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right]w=[3, 9, −12], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}5u−21​v+32​w.

Vector Operations

Practice: Collinear Vectors
Given  u⃗=⟨2,−5,6⟩,  v⃗=⟨−4,10,−12⟩,\vec{u}=\lang 2,−5,6\rang ,\ \ \vec{v}=\lang−4,10,−12\rang ,u=⟨2,−5,6⟩,  v=⟨−4,10,−12⟩, and  𝑤⃗=⟨−2,5,−6⟩𝑤⃗=\lang −2,5,−6 \rangw⃗=⟨−2,5,−6⟩, select all statements that are true.

Vector Operations: Linear Combinations

Find all values of kkk such that the vectors u=(−1,2,k,0)\bm{u}=(-1, 2,k,0)u=(−1,2,k,0) and v=(k,−6,6,0)\bm{v}=(k,-6,6,0)v=(k,−6,6,0) are collinear.

Vector Operations

Related Topics

Vector Operations and Linear Combinations

Practice: Collinear Vectors

More Vector Operations and Linear Combinations Questions:

Practice Question

Linearly independent vectors

Practice: Linear Combination

133 - FML 3 - 18.1W e.g. 23

Practice: Vector Operations

Practice: Vector Operations

Linear combinations

Write 1 Vector as linear combination of others

133 - FML 3 - 18.1W e.g. 49

133 - FML 3 - 18.1W e.g. 42

133 - FML 3 - 18.1W e.g. 45

19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{6}$_$\key{|Strc>5.2}$

Practice Question: When is 1 Vector a Linear Combination of Others (Has a Solution)

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 49

Practice Question: When is 1 Vector a Linear Combination of Others

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 45

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 42

19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{4}$_$\key{|Strc>5.2}$

19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{2}$_$\key{|Strc>5.2}$

Practice: Properties of Vector Operations

Vector Addition and Scalar Multiplication

Vector Operations

Practice: Vector Operations

Vector Operations

Vector Operations: Linear Combinations