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Vector Norm
Related Topics
Wize University Linear Algebra Textbook > Vectors
Length of a Vector (Vector Norm)
5 Activities
Practice: Properties of the Norm
Consider
u
⃗
,
v
⃗
,
w
⃗
∈
R
n
\vec{u},\ \vec{v},\ \vec{w}\ \in \reals^n
u
,
v
,
w
∈
R
n
. Select all of the statements that are
always true
.
A)
∣
∣
v
⃗
−
u
⃗
∣
∣
=
−
∣
∣
u
⃗
−
v
⃗
∣
∣
\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|
∣
∣
v
−
u
∣
∣
=
−
∣
∣
u
−
v
∣
∣
B)
∣
∣
u
⃗
∣
∣
+
∣
∣
v
⃗
∣
∣
=
∣
∣
u
⃗
+
v
⃗
∣
∣
\left|\left|\vec{u}\right|\right|+\left|\left|\vec{v}\right|\right|=\left|\left|\vec{u}+\vec{v}\right|\right|
∣
∣
u
∣
∣
+
∣
∣
v
∣
∣
=
∣
∣
u
+
v
∣
∣
C)
d
(
u
⃗
,
−
v
⃗
)
=
d
(
−
u
⃗
,
v
⃗
)
d\left(\vec{u},\ -\vec{v}\right)=d\left(-\vec{u},\ \vec{v}\right)
d
(
u
,
−
v
)
=
d
(
−
u
,
v
)
D)
∥
u
⃗
+
v
⃗
∥
=
∥
(
−
u
⃗
)
+
(
−
v
⃗
)
∥
\lVert \vec u + \vec v \rVert = \lVert (-\vec u) + (-\vec v) \rVert
∥
u
+
v
∥
=
∥(
−
u
)
+
(
−
v
)∥
I don't know
Check Submission
More Length of a Vector (Vector Norm) Questions:
Write the vectors
𝑢
⃗
=
(
1
,
2
,
3
)
𝑢⃗=(1,2,3)
u
⃗
=
(
1
,
2
,
3
)
,
𝑣
⃗
=
(
5
,
−
2
,
2
)
𝑣⃗=(\sqrt{5},−2,2)
v
⃗
=
(
5
,
−
2
,
2
)
, and
𝑤
⃗
=
(
3
,
3
,
3
)
\ 𝑤⃗=(\sqrt{3},\sqrt{3},3)
w
⃗
=
(
3
,
3
,
3
)
in increasing order of length.
(i.e. shortest vector first)
Vectors and coordinates (Components of a vector)
In diagram below two vectors
A
⃗
\vec{A}
A
and
B
⃗
\vec{B}
B
have shown. Vectors
C
⃗
\vec{C}
C
and
D
⃗
\vec{D}
D
are the sum
(
A
⃗
+
B
⃗
)
\left(\vec{A}+\vec{B}\right)
(
A
+
B
)
and difference
(
A
⃗
−
B
⃗
)
\left(\vec{A}-\vec{B}\right)
(
A
−
B
)
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
u
⃗
,
v
⃗
,
w
⃗
∈
R
3
\vec{u},\ \vec{v},\ \vec{w}\ \in R^3
u
,
v
,
w
∈
R
3
, which of the following statements is/are always true?
i.)
∣
∣
v
⃗
−
u
⃗
∣
∣
=
−
∣
∣
u
⃗
−
v
⃗
∣
∣
\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|
∣
∣
v
−
u
∣
∣
=
−
∣
∣
u
−
v
∣
∣
Practice: Magnitude of a 3D Vector
Practice: Magnitude of a 3D Vector
An aircraft's location is 200 miles S30
o
W 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
∣
u
⃗
∣
=
2
\bcb{|{\vec{u}|} = 2}
∣
u
∣
=
2
and
∣
v
⃗
∣
=
3
\bcb{|{ \vec{v} |} = 3}
∣
v
∣
=
3
, where
u
⃗
⋅
v
⃗
=
2
\bcb{\vec{u} \cdot \vec{v} = 2}
u
⋅
v
=
2
, find
∣
u
⃗
−
v
⃗
∣
\bcb{|{\vec{u} - \vec{v} }|}
∣
u
−
v
∣
.
133 - FML 3 - 18.1W e.g. 25
Given the vector
v
⃗
=
<
v
1
,
v
2
,
v
3
>
\bcb{\vec{v} = \left<v_1, v_2, v_3 \right>}
v
=
⟨
v
1
,
v
2
,
v
3
⟩
, find an expression for its length in terms of its components
v
1
,
v
2
,
\bcb{v_1,\, v_2,}
v
1
,
v
2
,
and
v
3
\bcb{v_3}
v
3
.
Length / magnitude / norm. 19.1W
1. Find the vector
d
⃗
\vd
d
connecting the points
P
1
(
−
2
,
−
1
)
P_1(-2,-1)
P
1
(
−
2
,
−
1
)
and
P
2
(
3
,
11
)
P_2(3,11)
P
2
(
3
,
11
)
.
2. Find the length of the vector
d
⃗
\vd
d
found, above. What does this length represent?
Given that the length of
𝑣
⃗
=
(
−
3
,
𝑘
)
\vec{𝑣}=(−3,𝑘)
v
=
(
−
3
,
k
)
is 5, find all possible values of 𝑘.
133 - FML 3 - 18.1W e.g. 25
Given the vector
v
⃗
=
<
v
1
,
v
2
,
v
3
>
\bcb{\vec{v} = \left<v_1, v_2, v_3 \right>}
v
=
⟨
v
1
,
v
2
,
v
3
⟩
, find an expression for its length in terms of its components
v
1
,
v
2
,
\bcb{v_1,\, v_2,}
v
1
,
v
2
,
and
v
3
\bcb{v_3}
v
3
.
Practice Question: Properties of Vector Operations
Practice Question: Properties of Vector Operations
If
u
⃗
,
v
⃗
,
w
⃗
∈
R
3
\vec{u},\ \vec{v},\ \vec{w}\ \in R^3
u
,
v
,
w
∈
R
3
, which of the following statements is/are always true?
i.)
∣
∣
v
⃗
−
u
⃗
∣
∣
=
−
∣
∣
u
⃗
−
v
⃗
∣
∣
\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|
∣
∣
v
−
u
∣
∣
=
−
∣
∣
u
−
v
∣
∣
Practice Question: Vector Operations and Length
Practice Question: Properties of Vector Operations
If
u
⃗
,
v
⃗
,
w
⃗
∈
R
3
\vec{u},\ \vec{v},\ \vec{w}\ \in R^3
u
,
v
,
w
∈
R
3
, which of the following statements is/are always true?
i.)
∣
∣
v
⃗
−
u
⃗
∣
∣
=
−
∣
∣
u
⃗
−
v
⃗
∣
∣
\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|
∣
∣
v
−
u
∣
∣
=
−
∣
∣
u
−
v
∣
∣
Practice: Norm of a Vector
Which of the following statements are true:
There exist vectors such that
∥
u
⃗
+
v
⃗
∥
=
∥
u
⃗
∥
+
∥
v
⃗
∥
\|\vec{u}+\vec{v}\|=\|\vec{u}\|+\|\vec{v}\|
∥
u
+
v
∥
=
∥
u
∥
+
∥
v
∥
If
∥
u
⃗
∥
=
∥
v
⃗
∥
\|\vec{u}\|=\|\vec{v}\|
∥
u
∥
=
∥
v
∥
then
u
⃗
=
v
⃗
\vec{u}=\vec{v}
u
=
v
Vector Norm
Practice: Distance Between Two Points/Vectors
Vector Norm
Practice: Vector Length
Given
v
⃗
=
[
−
3
k
]
\vec{v}= \begin{bmatrix} -3\\ k\\ \end{bmatrix}
v
=
[
−
3
k
]
and
∥
v
⃗
∥
=
5
\lVert \vec v \rVert = 5
∥
v
∥
=
5
, find all possible values of
k
k
k
.
Practice Question: Vector Length
Practice Question: Vector Length
Given that the length of
𝑣
⃗
=
(
−
3
,
𝑘
)
\vec{𝑣}=(−3,𝑘)
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
𝑣
⃗
=
(
−
3
,
𝑘
)
\vec{𝑣}=(−3,𝑘)
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
P
1
(
2
,
−
3
,
7
)
P_1(2,-3,7)
P
1
(
2
,
−
3
,
7
)
and
P
2
(
4
,
3
,
16
)
P_2(4,3,16)
P
2
(
4
,
3
,
16
)
is:
Dot and Cross Products: Vector Norm
If
u
⃗
,
v
⃗
,
w
⃗
∈
R
3
\vec{u},\ \vec{v},\ \vec{w}\ \ \in\ R^3
u
,
v
,
w
∈
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
∥
v
∥
where
v
=
(
2
,
3
,
−
3
,
−
4
,
0
)
\bm{v}=(\sqrt 2,3,-3,-4,0)
v
=
(
2
,
3
,
−
3
,
−
4
,
0
)