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Practice Question: Vector Properties
Related Topics
Wize University Linear Algebra Textbook > Vectors
Vector Properties
4 Activities
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?
i.)
(
−
u
⃗
)
⋅
v
⃗
=
(
−
v
⃗
)
⋅
u
⃗
(-\vec{u}) \cdot \vec{v} = (-\vec{v}) \cdot \vec{u}
(
−
u
)
⋅
v
=
(
−
v
)
⋅
u
ii.)
∣
∣
u
⃗
+
v
⃗
∣
∣
=
∣
∣
u
⃗
∣
∣
+
∣
∣
v
⃗
∣
∣
||\vec{u}+\vec{v}|| = ||\vec{u}|| + ||\vec{v}||
∣∣
u
+
v
∣∣
=
∣∣
u
∣∣
+
∣∣
v
∣∣
iii.)
u
⃗
×
v
⃗
=
v
⃗
×
u
⃗
\vec{u} \times \vec{v} = \vec{v} \times \vec{u}
u
×
v
=
v
×
u
iv.)
(
0
⃗
⋅
u
⃗
)
(
0
⃗
×
v
⃗
)
=
0
(\vec{0} \cdot \vec{u})(\vec{0} \times \vec{v}) = 0
(
0
⋅
u
)
(
0
×
v
)
=
0
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
.
$\tkct{cut from 19.4F}$ 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
.
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: 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