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Practice Question: Properties of Vector Operations

Related Topics

Wize University Linear Algebra Textbook > Vectors

Length of a Vector (Vector Norm)

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Practice Question: 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|
ii.) ∣∣u⃗∣∣+∣∣v⃗∣∣=∣∣u⃗+v⃗∣∣\left|\left|\vec{u}\right|\right|+\left|\left|\vec{v}\right|\right|=\left|\left|\vec{u}+\vec{v}\right|\right|
iii.) d(u⃗, −v⃗)=d(−u⃗, v⃗)d\left(\vec{u},\ -\vec{v}\right)=d\left(-\vec{u},\ \vec{v}\right)
iv.) 2v⃗+3u⃗=3v⃗+2u⃗2\vec{v}+3\vec{u}=3\vec{v}+2\vec{u}


More Length of a Vector (Vector Norm) Questions:
Write the vectors𝑢⃗=(1,2,3)𝑢⃗=(1,2,3), 𝑣⃗=(5,−2,2)𝑣⃗=(\sqrt{5},−2,2), and  𝑤⃗=(3,3,3)\ 𝑤⃗=(\sqrt{3},\sqrt{3},3) in increasing order of length.
(i.e. shortest vector first)
Vectors and coordinates (Components of a vector)
In diagram below two vectors A⃗\vec{A}and B⃗\vec{B} have shown. VectorsC⃗\vec{C}and D⃗\vec{D}are the sum (A⃗+B⃗)\left(\vec{A}+\vec{B}\right) and difference (A⃗−B⃗)\left(\vec{A}-\vec{B}\right)vectors, respectively.
Which statement is correct about the magnitude of four vectors?
Practice Question: Properties of Vector Operations
Practice Question: 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|
Practice: Magnitude of a 3D Vector
Practice: Magnitude of a 3D Vector
An aircraft's location is 200 miles S30oW from base, and it's altitude is 35,000 ft. Find the distance between the aircraft and the base.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | $\tkco{duplicate to mock ✓}$ 133 - FML 3 - 18.1W e.g. 46_$\tkcth{Mock F}\tkct{?}$_$\tkct{vid soln only}$
If ∣u⃗∣=2\bcb{|{\vec{u}|} = 2} and ∣v⃗∣=3\bcb{|{ \vec{v} |} = 3}, where u⃗⋅v⃗=2\bcb{\vec{u} \cdot \vec{v} = 2}, find ∣u⃗−v⃗∣\bcb{|{\vec{u} - \vec{v} }|}.
133 - FML 3 - 18.1W e.g. 25
Given the vector v⃗=<v1,v2,v3>\bcb{\vec{v} = \left<v_1, v_2, v_3 \right>}, find an expression for its length in terms of its components v1, v2,\bcb{v_1,\, v_2,} and v3\bcb{v_3}.
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?
Given that the length of 𝑣⃗=(−3,𝑘)\vec{𝑣}=(−3,𝑘) is 5, find all possible values of 𝑘.
133 - FML 3 - 18.1W e.g. 25
Given the vector v⃗=<v1,v2,v3>\bcb{\vec{v} = \left<v_1, v_2, v_3 \right>}, find an expression for its length in terms of its components v1, v2,\bcb{v_1,\, v_2,} and v3\bcb{v_3}.
Practice Question: Vector Operations and Length
Practice Question: 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|
Practice: Norm of a Vector
Which of the following statements are true:
There exist vectors such that ∥u⃗+v⃗∥=∥u⃗∥+∥v⃗∥\|\vec{u}+\vec{v}\|=\|\vec{u}\|+\|\vec{v}\|
If ∥u⃗∥=∥v⃗∥\|\vec{u}\|=\|\vec{v}\| then u⃗=v⃗\vec{u}=\vec{v}
Vector Norm
Practice: Properties of the Norm
Consider u⃗, v⃗, w⃗ ∈Rn\vec{u},\ \vec{v},\ \vec{w}\ \in \reals^n. Select all of the statements that are always true.
Vector Norm
Practice: Distance Between Two Points/Vectors
Vector Norm
Practice: Vector Length
Given v⃗=[−3k]\vec{v}= \begin{bmatrix} -3\\ k\\ \end{bmatrix} and ∥v⃗∥=5\lVert \vec v \rVert = 5 , find all possible values of kk.
Practice Question: Vector Length
Practice Question: Vector Length
Given that the length of 𝑣⃗=(−3,𝑘)\vec{𝑣}=(−3,𝑘) is 5, find all possible values of 𝑘.
Length of a Vector

Practice: Vector Length
Practice: Vector Length
Given that the length of 𝑣⃗=(−3,𝑘)\vec{𝑣}=(−3,𝑘) is 5, find all possible values of 𝑘.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |
The distance between the points P1(2,−3,7)P_1(2,-3,7) and P2(4,3,16)P_2(4,3,16) is:
Dot and Cross Products: Vector Norm
If u⃗, v⃗, w⃗  ∈ R3\vec{u},\ \vec{v},\ \vec{w}\ \ \in\ R^3, which of the following operations is/are defined?
(i.e. which of the following can be calculated?)
Select all that are correct.
Vector Norm: Length of a Vector
Find ∥v∥\lVert \bm{v} \rVert where v=(2,3,−3,−4,0)\bm{v}=(\sqrt 2,3,-3,-4,0)