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Practice Question: Dot and Cross Product Properties
Related Topics
Wize University Linear Algebra Textbook > Products of Vectors
Cross Product Properties
5 Activities
Practice Question: Dot and Cross Product Properties
Given that
u
⃗
=
(
1
,
−
2
,
0
,
1
)
\vec{u}=\left(1,-2,0,1\right)
u
=
(
1
,
−
2
,
0
,
1
)
and
𝑣
⃗
\vec𝑣
v
=
(
0
,
1
,
2
,
1
)
,
=(0,1,2,1),
=
(
0
,
1
,
2
,
1
)
,
which of the following is true?
i)
𝑢
⃗
×
𝑣
⃗
=
(
1
,
−
4
,
−
2
,
−
1
)
𝑢⃗ × \vec𝑣 = (1, −4, −2, −1)
u
⃗
×
v
=
(
1
,
−
4
,
−
2
,
−
1
)
ii)
∣
u
⃗
∙
v
⃗
∣
=
1
\left|\vec{u}\ \bullet\ \vec{v}\right|=1
∣
u
∙
v
∣
=
1
iii)
‖
𝑢
⃗
+
𝑣
⃗
‖
=
10
‖𝑢⃗ + \vec𝑣 ‖ = \sqrt{10}
‖
u
⃗
+
v
‖
=
10
I don't know
Check Submission
More Cross Product Properties Questions:
𝑢
⃗
,
𝑣
⃗
,
𝑤
⃗
𝑢⃗ ,\vec{𝑣} , \vec𝑤
u
⃗
,
v
,
w
are vectors in
R
3
ℝ^3
R
3
and 𝑐, 𝑑 are scalars. Which of the following produce(s) a vector?
i)
(
𝑤
⃗
×
𝑐
𝑢
⃗
)
+
𝑑
𝑣
⃗
(\vec𝑤× 𝑐𝑢⃗ ) + 𝑑\vec𝑣
(
w
×
c
u
⃗
)
+
d
v
ii)
(
w
⃗
×
𝑐
𝑢
⃗
)
⋅
𝑑
𝑣
(\vec{w}×𝑐𝑢⃗ ) \cdot 𝑑𝑣
(
w
×
c
u
⃗
)
⋅
d
v
Practice Question: Properties of Dot and Cross Products
Practice Question: Properties of Dot and Cross Products
𝑢
⃗
,
𝑣
⃗
,
𝑤
⃗
𝑢⃗ ,\vec{𝑣} , \vec𝑤
u
⃗
,
v
,
w
are vectors in
R
3
ℝ^3
R
3
and 𝑐, 𝑑 are scalars. Which of the following produce(s) a vector?
i)
(
𝑤
⃗
×
𝑐
𝑢
⃗
)
+
𝑑
𝑣
⃗
(\vec𝑤× 𝑐𝑢⃗ ) + 𝑑\vec𝑣
(
w
×
c
u
⃗
)
+
d
v
Practice Question: Vector Properties
Practice Question: Vector Properties
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?)
Practice: Dot and Cross Product Properties
Practice: Dot and Cross Product Properties
𝑢
⃗
,
𝑣
⃗
,
𝑤
⃗
𝑢⃗ , \vec𝑣 , 𝑤⃗
u
⃗
,
v
,
w
⃗
are vectors in
R
3
ℝ^3
R
3
and 𝑐, 𝑑 are scalars. Which of the following is/are true?
Practice: Cross Product (1)
Do the vectors
u
⃗
=
<
3
,
−
1
,
4
>
,
v
⃗
=
<
6
,
0
,
1
>
,
and
w
⃗
=
<
−
1
,
2
,
5
>
\vec{u}=~<3,~-1,~4>,~\vec{v}=~<6,~0,~1>,~\text{and}~\vec{w}=~<-1,~2,~5>~
u
=
<
3
,
−
1
,
4
>
,
v
=
<
6
,
0
,
1
>
,
and
w
=
<
−
1
,
2
,
5
>
lie in the same plane?
Cross Product Properties
Practice: Cross Product Properties
Let
u
⃗
,
v
⃗
,
w
⃗
∈
R
3
\vec u ,\ \vec{v} ,\ \vec w \in \reals^3
u
,
v
,
w
∈
R
3
be vectors.
Let
c
,
d
∈
R
c,\ d \in \reals
c
,
d
∈
R
be scalars.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 51_$\tkcth{Mock F}\tkct{?}$_$\tkct{no soln / vid}$
If
u
⃗
=
<
1
,
−
2
,
3
>
\bcb{\vec{u} = \left< 1, -2, 3 \right>}
u
=
⟨
1
,
−
2
,
3
⟩
and
v
⃗
=
<
2
,
−
1
,
3
>
\bcb{\vec{v} = \left< 2, -1, 3 \right>}
v
=
⟨
2
,
−
1
,
3
⟩
then
u
⃗
×
v
⃗
\bcb{\vec{u} \times \vec{v}}
u
×
v
is parallel to:
Additional practice problems--vector products
Additional Practice Problems--Vector Products
1. If
u
⃗
=
(
−
2
,
3
,
−
1
)
,
v
⃗
=
(
−
1
,
−
2
,
3
)
\vec{u}=\left(-2,3,-1\right),\ \ \vec{v}=\left(-1,-2,3\right)
u
=
(
−
2
,
3
,
−
1
)
,
v
=
(
−
1
,
−
2
,
3
)
, find
a)
u
⃗
∙
v
⃗
\vec{u}\ \bullet\ \vec{v}
u
∙
v
Given that
u
→
\overrightarrow{u}
u
,
v
→
\overrightarrow{v}
v
and
w
→
\overrightarrow{w}
w
are vectors in
R
3
\mathbb{R}^3
R
3
and
k
k
k
is a scalar (constant). Which of the following operations are defined (possible)?
u
→
⋅
(
v
→
×
w
→
)
\overrightarrow{u}\cdot(\overrightarrow{v}\times\overrightarrow{w})
u
⋅
(
v
×
w
)
u
→
+
(
v
→
⋅
w
→
)
\overrightarrow{u}+(\overrightarrow{v}\cdot\overrightarrow{w})
u
+
(
v
⋅
w
)