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Dot Product Properties
Related Topics
Wize University Linear Algebra Textbook > Products of Vectors
Dot Product Properties
5 Activities
Practice: Dot Product Properties
Let
u
⃗
,
v
⃗
,
w
⃗
∈
R
3
\vec{u},\ \vec{v}, \vec{w} \in \reals^3
u
,
v
,
w
∈
R
3
be vectors and let
c
,
d
∈
R
c,\ d \in \reals
c
,
d
∈
R
be scalars. Select all expressions that are
valid
.
A)
c
u
⃗
+
d
(
v
⃗
⋅
w
⃗
)
c\vec{u}+d(\vec{v}\cdot \vec{w})
c
u
+
d
(
v
⋅
w
)
B)
c
u
⃗
+
(
d
v
⃗
⋅
w
⃗
)
c\vec{u}+\left(d\vec{v}\cdot\vec{w}\right)
c
u
+
(
d
v
⋅
w
)
C)
(
c
u
⃗
+
d
v
⃗
)
⋅
w
⃗
\left(c\vec{u}+d\vec{v}\right)\cdot \vec{w}
(
c
u
+
d
v
)
⋅
w
D)
c
(
u
⃗
+
d
v
⃗
)
⋅
w
⃗
c\left(\vec{u}+d\vec{v}\right)\cdot \vec{w}
c
(
u
+
d
v
)
⋅
w
I don't know
Check Submission
More Dot Product Properties Questions:
133 - FML 3 - 18.1W e.g. 46
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. 40
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
⟩
, find
(
2
u
⃗
)
⋅
(
3
v
⃗
)
\bcb{(2\vec{u})\cdot (3\vec{v})}
(
2
u
)
⋅
(
3
v
)
.
133 - FML 3 - 18.1W e.g. 22
How do you express the idea that two vectors
v
⃗
\bcb{\vec{v}}
v
and
w
⃗
\bcb{\vec{w}}
w
are perpendicular, mathematically?
Practice: Dot Product Properties
Practice: Dot Product Properties
Suppose that
u
⃗
,
v
⃗
,
w
⃗
\vec{u},\ \vec{v},\ \vec{w}
u
,
v
,
w
are vectors in
R
2
R^2
R
2
,
p
⃗
,
q
⃗
,
r
⃗
\vec{p},\ \vec{q},\ \vec{r}
p
,
q
,
r
are vectors in
R
3
R^3
R
3
and
a
,
b
a,\ b
a
,
b
are scalars.
Which of the following are valid operations? (Select all that apply)
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?
133 - FML 3 - 18.1W e.g. 54
Find the value of
k
\bcb{k}
k
such that the vector
u
⃗
=
<
1
,
k
,
3
>
\bcb{\vec{u} = \left< 1, k, 3 \right>}
u
=
⟨
1
,
k
,
3
⟩
is perpendicular to the vector
v
⃗
=
<
2
,
−
1
,
k
>
\bcb{\vec{v} = \left< 2, -1, k\right>}
v
=
⟨
2
,
−
1
,
k
⟩
.
Practice: Dot Product Properties
Practice: Dot Product Properties
Suppose that
u
⃗
,
v
⃗
,
w
⃗
\vec{u},\ \vec{v},\ \vec{w}
u
,
v
,
w
are vectors in
R
2
R^2
R
2
,
p
⃗
,
q
⃗
,
r
⃗
\vec{p},\ \vec{q},\ \vec{r}
p
,
q
,
r
are vectors in
R
3
R^3
R
3
and
a
,
b
a,\ b
a
,
b
are scalars.
Which of the following are valid operations? (Select all that apply)
Find the value of 𝑘 such that the vectors
𝑢
⃗
=
(
1
,
𝑘
,
2
𝑘
)
𝑢⃗=(1,𝑘,2𝑘)
u
⃗
=
(
1
,
k
,
2
k
)
𝑎𝑛𝑑
𝑣
⃗
=
(
3
,
1
,
0
)
𝑣⃗=(3,1,0)
v
⃗
=
(
3
,
1
,
0
)
are orthogonal.
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
→
=
(
5
,
−
1
,
2
,
k
)
\overrightarrow{u}=\left(5,-1,2,k\right)
u
=
(
5
,
−
1
,
2
,
k
)
and
v
→
=
(
−
10
,
2
,
−
4
,
6
)
\overrightarrow{v}=\left(-10,2,-4,6\right)
v
=
(
−
10
,
2
,
−
4
,
6
)
, which of the following statements is/are true?
If
k
=
3
k=3
k
=
3
, then
u
→
\overrightarrow{u}
u
and
v
→
\overrightarrow{v}
v
are parallel/collinear
If
k
=
−
10
k=-10
k
=
−
10
, then
u
→
\overrightarrow{u}
u
and
v
→
\overrightarrow{v}
v
are orthogonal/perpendicular
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
)
Dot Product Properties
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 orthogonal.