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Length / magnitude / norm. 19.1W
Related Topics
Wize University Linear Algebra Textbook > Vectors
Vector Properties
4 Activities
Wize University Linear Algebra Textbook > Vectors
Length of a Vector (Vector Norm)
5 Activities
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?
Part 1
Part 2
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
)
.
<
5
,
12
>
\rowvec{5}{12}
⟨
5
,
12
⟩
<
−
5
,
−
12
>
\rowvec{-5}{-12}
⟨
−
5
,
−
12
⟩
<
1
,
10
>
\rowvec{1}{10}
⟨
1
,
10
⟩
<
−
1
,
−
10
>
\rowvec{-1}{-10}
⟨
−
1
,
−
10
⟩
I don't know
Previous Part
Check Part 1
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Check Submission
More Vector Properties Questions:
Practice: Vector Addition and Subtraction
The coordinates of four corners of a Parallelogram
A
B
C
D
ABCD
A
B
C
D
in a Clock-Wise order are
A
=
(
1
,
3
)
A=(1,3)
A
=
(
1
,
3
)
,
B
=
(
−
1
,
5
)
B=(-1,5)
B
=
(
−
1
,
5
)
,
C
C
C
and
D
=
(
4
,
7
)
D=(4,7)
D
=
(
4
,
7
)
. Find missing coordinates of
C
C
C
.
$\tkct{cut from 19.4F}$ 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
u
⃗
=
P
Q
⃗
\bcb{\ol{u} = \vec{PQ}}
u
=
PQ
, find a vector parallel but in the opposite direction to
u
⃗
\bcb{\vec{u}}
u
.
A parallelogram in
R
3
R^3
R
3
has vertices
A
(
−
1
,
1
,
2
)
A\left(-1,1,2\right)
A
(
−
1
,
1
,
2
)
,
B
(
1
,
−
2
,
8
)
B\left(1,-2,8\right)
B
(
1
,
−
2
,
8
)
and
C
=
(
3
,
−
1
,
2
)
C=\left(3,-1,2\right)
C
=
(
3
,
−
1
,
2
)
.
Vector Arithmetic
Suppose
X
=
(
x
1
,
x
2
,
x
3
)
X=(x_1, x_2, x_3)
X
=
(
x
1
,
x
2
,
x
3
)
and
Y
=
(
y
1
,
y
2
,
y
3
)
Y=(y_1, y_2, y_3)
Y
=
(
y
1
,
y
2
,
y
3
)
If
−
X
Y
⃗
=
<
4
3
,
−
3
5
,
10
3
>
\vec{-XY}=\left<\frac{4}{3},\frac{-3}{5}, \frac{10}{3}\right>
−
X
Y
=
⟨
3
4
,
5
−
3
,
3
10
⟩
,
what is
Y
X
⃗
\vec{YX}
Y
X
?
Practice Question 9: Vector Operations
Practice Question: Vector Operations
Let
u
⃗
=
(
−
9
,
6
)
\vec{u}=\left(-9,\ 6\right)
u
=
(
−
9
,
6
)
and
v
⃗
=
(
1
,
−
1
)
\vec{v}=\left(1,-1\right)
v
=
(
1
,
−
1
)
. If we start at the origin and travel along a directed line segment that is opposite to
u
⃗
\vec{u}
u
but has 1/3 of its length, then we travel along the translation of
v
⃗
\vec{v}
v
to your current position. Finally, we travel along the translation of the unit vector that points in the same direction as
v
⃗
\vec{v}
v
.
a.) Where do we end up?
If
α
∈
R
\alpha \in \mathbb{R}
α
∈
R
and
v
∈
V
v \in V
v
∈
V
, where
V
V
V
is a real vector space and
α
v
=
0
\alpha v = 0
α
v
=
0
, then prove that either
α
=
0
\alpha = 0
α
=
0
or
v
=
0
v = 0
v
=
0
.
Practice: Vector Addition and Subtraction
The coordinates of four corners of a Parallelogram
A
B
C
D
ABCD
A
B
C
D
in a Clock-Wise order are
A
=
(
1
,
3
)
A=(1,3)
A
=
(
1
,
3
)
,
B
=
(
−
1
,
5
)
B=(-1,5)
B
=
(
−
1
,
5
)
,
C
C
C
and
D
=
(
4
,
7
)
D=(4,7)
D
=
(
4
,
7
)
. Find missing coordinates of
C
C
C
.
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
u
⃗
\vec{u}
u
and
v
⃗
\vec{v}
v
are vectors in
R
3
\mathbb{R}^3
R
3
, which of the following statements is/are
always
true?
Practice: Vector Operations
A parallelogram has sides
A
B
AB
A
B
,
B
C
BC
B
C
,
C
D
CD
C
D
and
D
A
DA
D
A
The following coordinates are known:
A
(
2
,
0
,
1
)
A(2,0,1)
A
(
2
,
0
,
1
)
,
C
(
6
,
0
,
0
)
C(6,0,0)
C
(
6
,
0
,
0
)
, and
M
(
3
,
1
,
0
)
M(3,1,0)
M
(
3
,
1
,
0
)
, where
M
M
M
is the midpoint between
A
A
A
and
B
B
B
Find the vector
B
D
⃗
\vec{BD}
B
D
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
.
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: 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
.
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
)