High School
SAT
ACT
University
MCAT
LSAT
DAT

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

Related Topics

Wize University Linear Algebra Textbook > Vectors

Vector Operations and Linear Combinations

7 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Find a vector parallel to u⃗=<1, −1, 1>\bcb{\vec{u} = \left< 1,\, -1,\, 1\right>}.
More Vector Operations and Linear Combinations Questions:
Practice Question
Is the vector (π,e,1)(\pi, e, 1) in the linear span of (1,2,3),(−1,0,1)(1, 2, 3), (-1, 0, 1) and (5,1,−2)(5, 1, -2) of R3\mathbb{R}^3 ? 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}^n is a linear combination of x⃗,y⃗∈Rn\vec x,\vec y \in \mathbb{R}^n and v⃗∈Rn\vec v \in \mathbb{R}^n is a (different) linear combination of the same x⃗,y⃗\vec x,\vec y then every linear combination au⃗+bv⃗a\vec u+b\vec v for a,b∈Ra,b\in\mathbb{R} is also a linear combination of x⃗,y⃗\vec x,\vec y .
133 - FML 3 - 18.1W e.g. 23
Give a mathematical expression that mean v⃗\bcb{\vec{v}} and w⃗\bcb{\vec{w}} are parallel.
Practice: Vector Operations
Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}.
Vector Operations
Practice: Collinear Vectors
Find the value of cc such that the vectors u⃗=⟨−3,c,3⟩\vec u = \lang −3,c,3 \rang and v⃗=⟨1,−2,−1⟩\vec v = \lang 1,−2,−1 \rang are collinear.
Practice: Vector Operations
Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}.
Linear combinations
If u⃗=[5−23]\vec{u}=\begin{bmatrix}5\\-2\\3\end{bmatrix}is a linear combination of v⃗=[−2k3]\vec{v}=\begin{bmatrix}-2\\k\\3\end{bmatrix}and w⃗=[32−1]\vec{w}=\begin{bmatrix}3\\2\\-1\end{bmatrix}, what is the value of kk?
Write 1 Vector as linear combination of others
Write (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), 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>}, find a unit vector perpendicular to v⃗\bcb{\vec{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>}.
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}}.
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} and w⃗=[−121]\vw = \colvecth{-1}{2}{1}.
Practice Question: When is 1 Vector a Linear Combination of Others (Has a Solution)
Write (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) , 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>}, find a unit vector perpendicular to v⃗\bcb{\vec{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) 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)? If yes write down u⃗\vec{u}as 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)} 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}}.
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}.
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}.
Practice: Properties of Vector Operations
Practice: Properties of Vector Operations
If u⃗, v⃗, w⃗ ∈R3\vec{u},\ \vec{v},\ \vec{w}\ \in R^3, 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|
Vector Addition and Scalar Multiplication

Vector Operations
Practice: Collinear Vectors
Find the value of cc such that the vectors u⃗=⟨−3,c,3⟩\vec u = \lang −3,c,3 \rang and v⃗=⟨1,−2,−1⟩\vec v = \lang 1,−2,−1 \rang are collinear.
Practice: Vector Operations
Practice: Vector Operations
If u⃗=[1, 0, −1]\vec{u}=\left[1,\ 0,\ -1\right], v⃗=[−2, 4, 1]\vec{v}=\left[-2,\ 4,\ 1\right], and w⃗=[3, 9, −12]\vec{w}=\left[3,\ 9,\ -12\right], determine 5u⃗−12v⃗+23w⃗5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{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 , and 𝑤⃗=⟨−2,5,−6⟩𝑤⃗=\lang −2,5,−6 \rang, select all statements that are true.
Vector Operations: Linear Combinations
Find all values of kk such that the vectors u=(−1,2,k,0)\bm{u}=(-1, 2,k,0) and v=(k,−6,6,0)\bm{v}=(k,-6,6,0) are collinear.