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$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 45
Related Topics
Wize University Linear Algebra Textbook > Vectors
Vector Properties
4 Activities
Wize University Linear Algebra Textbook > Vectors
Vector Operations and Linear Combinations
7 Activities
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
.
<
−
4
,
−
2
,
3
>
\left< -4, -2, 3\right>
⟨
−
4
,
−
2
,
3
⟩
<
2
,
−
4
,
3
>
\left< 2, -4, 3\right>
⟨
2
,
−
4
,
3
⟩
<
4
,
2
,
−
3
>
\left< 4, 2, -3\right>
⟨
4
,
2
,
−
3
⟩
<
2
,
−
3
,
4
>
\left< 2, -3, 4\right>
⟨
2
,
−
3
,
4
⟩
I don't know
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
.
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?
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 Vector Operations and Linear Combinations Questions:
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
R
3
\mathbb{R}^3
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
u
⃗
∈
R
n
\vec u \in \mathbb{R}^n
u
∈
R
n
is a linear combination of
x
⃗
,
y
⃗
∈
R
n
\vec x,\vec y \in \mathbb{R}^n
x
,
y
∈
R
n
and
v
⃗
∈
R
n
\vec v \in \mathbb{R}^n
v
∈
R
n
is a (different) linear combination of the same
x
⃗
,
y
⃗
\vec x,\vec y
x
,
y
then every linear combination
a
u
⃗
+
b
v
⃗
a\vec u+b\vec v
a
u
+
b
v
for
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
is also a linear combination of
x
⃗
,
y
⃗
\vec x,\vec y
x
,
y
.
133 - FML 3 - 18.1W e.g. 23
Give a mathematical expression that mean
v
⃗
\bcb{\vec{v}}
v
and
w
⃗
\bcb{\vec{w}}
w
are parallel.
Practice: Vector Operations
Practice: Vector Operations
If
u
⃗
=
[
1
,
0
,
−
1
]
\vec{u}=\left[1,\ 0,\ -1\right]
u
=
[
1
,
0
,
−
1
]
,
v
⃗
=
[
−
2
,
4
,
1
]
\vec{v}=\left[-2,\ 4,\ 1\right]
v
=
[
−
2
,
4
,
1
]
, and
w
⃗
=
[
3
,
9
,
−
12
]
\vec{w}=\left[3,\ 9,\ -12\right]
w
=
[
3
,
9
,
−
12
]
, determine
5
u
⃗
−
1
2
v
⃗
+
2
3
w
⃗
5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}
5
u
−
2
1
v
+
3
2
w
.
Vector Operations
Practice: Collinear Vectors
Find the value of
c
c
c
such that the vectors
u
⃗
=
⟨
−
3
,
c
,
3
⟩
\vec u = \lang −3,c,3 \rang
u
=
⟨
−
3
,
c
,
3
⟩
and
v
⃗
=
⟨
1
,
−
2
,
−
1
⟩
\vec v = \lang 1,−2,−1 \rang
v
=
⟨
1
,
−
2
,
−
1
⟩
are collinear.
Practice: Vector Operations
Practice: Vector Operations
If
u
⃗
=
[
1
,
0
,
−
1
]
\vec{u}=\left[1,\ 0,\ -1\right]
u
=
[
1
,
0
,
−
1
]
,
v
⃗
=
[
−
2
,
4
,
1
]
\vec{v}=\left[-2,\ 4,\ 1\right]
v
=
[
−
2
,
4
,
1
]
, and
w
⃗
=
[
3
,
9
,
−
12
]
\vec{w}=\left[3,\ 9,\ -12\right]
w
=
[
3
,
9
,
−
12
]
, determine
5
u
⃗
−
1
2
v
⃗
+
2
3
w
⃗
5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}
5
u
−
2
1
v
+
3
2
w
.
Linear combinations
If
u
⃗
=
[
5
−
2
3
]
\vec{u}=\begin{bmatrix}5\\-2\\3\end{bmatrix}
u
=
5
−
2
3
is a linear combination of
v
⃗
=
[
−
2
k
3
]
\vec{v}=\begin{bmatrix}-2\\k\\3\end{bmatrix}
v
=
−
2
k
3
and
w
⃗
=
[
3
2
−
1
]
\vec{w}=\begin{bmatrix}3\\2\\-1\end{bmatrix}
w
=
3
2
−
1
, what is the value of
k
k
k
?
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
v
⃗
=
<
0
,
4
,
3
>
\bcb{\ol{v} = \left< 0,\, 4,\, 3 \right>}
v
=
⟨
0
,
4
,
3
⟩
, find a unit vector perpendicular to
v
⃗
\bcb{\vec{v}}
v
.
133 - FML 3 - 18.1W e.g. 42
Find a vector parallel to
u
⃗
=
<
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
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
.
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:
v
⃗
=
[
1
1
1
]
\vv = \colvecth{1}{1}{1}
v
=
1
1
1
and
w
⃗
=
[
−
1
2
1
]
\vw = \colvecth{-1}{2}{1}
w
=
−
1
2
1
.
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
v
⃗
=
<
0
,
4
,
3
>
\bcb{\ol{v} = \left< 0,\, 4,\, 3 \right>}
v
=
⟨
0
,
4
,
3
⟩
, find a unit vector perpendicular to
v
⃗
\bcb{\vec{v}}
v
.
Practice Question: When is 1 Vector a Linear Combination of Others
Is it possible to write down
u
⃗
=
(
1
,
2
,
3
)
\vec{u}=(1,2,3)
u
=
(
1
,
2
,
3
)
as a linear combination of
v
1
⃗
=
(
1
,
0
,
2
)
,
v
2
⃗
=
(
0
,
1
,
2
)
and
v
3
⃗
=
(
2
,
1
,
6
)
\vec{v_1}=(1,0,2),\vec{v_2}=(0,1,2)\text{ and }\vec{v_3}=(2,1,6)
v
1
=
(
1
,
0
,
2
)
,
v
2
=
(
0
,
1
,
2
)
and
v
3
=
(
2
,
1
,
6
)
? If yes write down
u
⃗
\vec{u}
u
as a linear combination of other vectors.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 42
Find a vector parallel to
u
⃗
=
<
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:
v
⃗
=
[
−
1
2
1
]
\vv = \colvecth{-1}{2}{1}
v
=
−
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
v
⃗
=
[
1
1
]
\vv = \colvec{1}{1}
v
=
[
1
1
]
.
Practice: Properties of Vector Operations
Practice: 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
∣
∣
Vector Addition and Scalar Multiplication
Vector Operations
Practice: Collinear Vectors
Find the value of
c
c
c
such that the vectors
u
⃗
=
⟨
−
3
,
c
,
3
⟩
\vec u = \lang −3,c,3 \rang
u
=
⟨
−
3
,
c
,
3
⟩
and
v
⃗
=
⟨
1
,
−
2
,
−
1
⟩
\vec v = \lang 1,−2,−1 \rang
v
=
⟨
1
,
−
2
,
−
1
⟩
are collinear.
Practice: Vector Operations
Practice: Vector Operations
If
u
⃗
=
[
1
,
0
,
−
1
]
\vec{u}=\left[1,\ 0,\ -1\right]
u
=
[
1
,
0
,
−
1
]
,
v
⃗
=
[
−
2
,
4
,
1
]
\vec{v}=\left[-2,\ 4,\ 1\right]
v
=
[
−
2
,
4
,
1
]
, and
w
⃗
=
[
3
,
9
,
−
12
]
\vec{w}=\left[3,\ 9,\ -12\right]
w
=
[
3
,
9
,
−
12
]
, determine
5
u
⃗
−
1
2
v
⃗
+
2
3
w
⃗
5\vec{u}-\frac{1}{2}\vec{v}+\frac{2}{3}\vec{w}
5
u
−
2
1
v
+
3
2
w
.
Vector Operations
Practice: Collinear Vectors
Given
u
⃗
=
⟨
2
,
−
5
,
6
⟩
,
v
⃗
=
⟨
−
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
𝑤
⃗
=
⟨
−
2
,
5
,
−
6
⟩
𝑤⃗=\lang −2,5,−6 \rang
w
⃗
=
⟨
−
2
,
5
,
−
6
⟩
, select all statements that are true.
Vector Operations: Linear Combinations
Find all values of
k
k
k
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.