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Give that 𝑢⃗ and 𝑣 are vectors in R^3, which of the following statements is/a…
Related Topics
Wize University Linear Algebra Textbook > Products of Vectors
Cross Product
3 Activities
Give that
𝑢
⃗
𝑢⃗
u
⃗
and
𝑣
⃗
\vec𝑣
v
are vectors in
R
3
\mathbb{R}^3
R
3
, which of the following statements is/are always true?
i.
2
𝑢
⃗
+
3
𝑣
⃗
=
2
𝑣
⃗
+
3
𝑢
⃗
2𝑢⃗ + 3\vec𝑣 = 2\vec𝑣 + 3𝑢⃗
2
u
⃗
+
3
v
=
2
v
+
3
u
⃗
ii.
2
𝑢
⃗
.
(
−
3
𝑣
⃗
)
=
−
2
(
3
𝑢
⃗
.
𝑣
⃗
)
2𝑢⃗ . (−3\vec𝑣 ) = −2(3𝑢⃗ . \vec𝑣 )
2
u
⃗.
(
−
3
v
)
=
−
2
(
3
u
⃗.
v
)
iii.
𝑢
⃗
×
(
−
3
𝑣
⃗
)
=
3
(
𝑣
⃗
×
𝑢
⃗
)
𝑢⃗ × (−3\vec𝑣 ) = 3(\vec𝑣 × 𝑢⃗ )
u
⃗
×
(
−
3
v
)
=
3
(
v
×
u
⃗
)
i. and ii. only
ii. only
iii. only
ii. and iii. only
None of the above
I don't know
Check Submission
More Cross Product Questions:
Cross Product
Practice: Orthogonal Vectors
Find a vector orthogonal to both
u
⃗
=
[
2
−
1
3
]
\vec u = \begin{bmatrix} 2\\ -1\\ 3 \end{bmatrix}
u
=
2
−
1
3
and
v
⃗
=
[
0
−
1
−
1
]
\vec{v} = \begin{bmatrix} 0\\ -1\\ -1\\ \end{bmatrix}
v
=
0
−
1
−
1
.
For extra practice, check your answer!
Find vec form line thru pt ortho to 2 vec form lines
Find the equation of the line that passes through the point
(
1
,
2
,
−
3
)
\left(1,2,-3\right)
(
1
,
2
,
−
3
)
and is orthogonal to the lines with equations:
(
x
,
y
,
z
)
=
(
−
5
,
0
,
1
)
+
t
(
1
,
0
,
−
1
)
(x,y,z)= (-5,0,1)+t(1,0,-1)
(
x
,
y
,
z
)
=
(
−
5
,
0
,
1
)
+
t
(
1
,
0
,
−
1
)
and
(
x
,
y
,
z
)
=
(
3
,
−
1
,
6
)
+
t
(
0
,
2
,
1
)
(x,y,z)=(3,-1,6)+t(0,2,1)
(
x
,
y
,
z
)
=
(
3
,
−
1
,
6
)
+
t
(
0
,
2
,
1
)
Which of the following vectors is orthogonal to both
[
1
,
2
,
3
]
[1,2,3]
[
1
,
2
,
3
]
and
[
−
3
,
1
,
0
]
[-3,1,0]
[
−
3
,
1
,
0
]
.
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find a vector orthogonal to both
𝑢
⃗
=
(
2
,
−
1
,
3
)
𝑢⃗=(2,−1,3)
u
⃗
=
(
2
,
−
1
,
3
)
and
𝑣
⃗
=
(
0
,
−
1
,
−
1
)
.
\vec{𝑣} = (0, −1, −1).
v
=
(
0
,
−
1
,
−
1
)
.
Consider three vectors
A
⃗
=
2
i
^
+
j
^
,
B
⃗
=
i
^
−
2
j
^
\vec{A}=2\hat{i}+\hat{j},\,\,\vec{B}=\hat{i}-2\hat{j}
A
=
2
i
^
+
j
^
,
B
=
i
^
−
2
j
^
, and
C
⃗
=
−
2
i
^
+
j
^
\vec{C}=-2\hat{i}+\hat{j}
C
=
−
2
i
^
+
j
^
. The angle between
C
⃗
\vec{C}
C
and
A
⃗
×
B
⃗
\vec{A}\times\vec{B}
A
×
B
is?
Evaluate
2
k
^
×
(
3
i
^
×
2
j
^
)
2\hat{k}\times(3\hat{i}\times2\hat{j})
2
k
^
×
(
3
i
^
×
2
j
^
)
Practice: Cross Product
Practice: Cross Product
Recall that
i
⃗
=
[
1
,
0
,
0
]
\vec{i}=\left[1,\ 0,\ 0\right]
i
=
[
1
,
0
,
0
]
,
j
⃗
=
[
0
,
1
,
0
]
\vec{j}=\left[0,\ 1,\ 0\right]
j
=
[
0
,
1
,
0
]
, and
k
⃗
=
[
0
,
0
,
1
]
\vec{k}=\left[0,\ 0,\ 1\right]
k
=
[
0
,
0
,
1
]
.
Match the following cross products with the correct result.
Cross Product
Practice: Orthogonal Vectors
Find a vector orthogonal to both
u
⃗
=
[
2
−
1
3
]
\vec u = \begin{bmatrix} 2\\ -1\\ 3 \end{bmatrix}
u
=
2
−
1
3
and
v
⃗
=
[
0
−
1
−
1
]
\vec{v} = \begin{bmatrix} 0\\ -1\\ -1\\ \end{bmatrix}
v
=
0
−
1
−
1
.
For extra practice, check your answer!
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find a vector orthogonal to both
𝑢
⃗
=
(
2
,
−
1
,
3
)
𝑢⃗=(2,−1,3)
u
⃗
=
(
2
,
−
1
,
3
)
and
𝑣
⃗
=
(
0
,
−
1
,
−
1
)
.
\vec{𝑣} = (0, −1, −1).
v
=
(
0
,
−
1
,
−
1
)
.
Practice: Orthogonal Vectors
Practice: Orthogonal Vectors
Find a vector orthogonal to both
𝑢
⃗
=
(
2
,
−
1
,
3
)
𝑢⃗=(2,−1,3)
u
⃗
=
(
2
,
−
1
,
3
)
and
𝑣
⃗
=
(
0
,
−
1
,
−
1
)
.
\vec{𝑣} = (0, −1, −1).
v
=
(
0
,
−
1
,
−
1
)
.
Practice: Cross Product
Practice: Cross Product
Consider
u
⃗
=
[
−
2
,
3
,
−
1
]
\vec{u}=\left[-2,\ 3,\ -1\right]
u
=
[
−
2
,
3
,
−
1
]
and
v
⃗
=
[
1
,
0
,
−
1
]
\vec{v}=\left[1,\ 0,\ -1\right]
v
=
[
1
,
0
,
−
1
]
.
If
u
⃗
=
(
2
,
0
,
−
3
)
,
v
⃗
=
(
−
2
,
2
,
−
4
)
\vec{u}=\left(2,0,-3\right),\ \vec{v}=\left(-2,2,-4\right)
u
=
(
2
,
0
,
−
3
)
,
v
=
(
−
2
,
2
,
−
4
)
and
w
⃗
=
(
1
,
2
,
1
)
\vec{w}=\left(1,2,1\right)
w
=
(
1
,
2
,
1
)
, find a vector that is orthogonal to
u
v
⃗
\vec{uv}
uv
and
u
w
⃗
\vec{uw}
u
w
.
If
𝑢
⃗
=
(
2
,
0
,
−
3
)
,
𝑣
⃗
=
(
−
2
,
2
,
−
4
)
,
𝑢⃗ = (2, 0, −3), \vec𝑣 = (−2, 2, −4),
u
⃗
=
(
2
,
0
,
−
3
)
,
v
=
(
−
2
,
2
,
−
4
)
,
and
𝑤
⃗
=
(
1
,
2
,
1
)
𝑤⃗= (1,2,1)
w
⃗
=
(
1
,
2
,
1
)
, compute
u
⃗
×
w
⃗
\vec{u}\ \times\ \vec{w}
u
×
w
.
If
𝑢
⃗
=
(
2
,
0
,
−
3
)
,
𝑣
⃗
=
(
−
2
,
2
,
−
4
)
,
𝑢⃗ = (2, 0, −3), \vec𝑣 = (−2, 2, −4),
u
⃗
=
(
2
,
0
,
−
3
)
,
v
=
(
−
2
,
2
,
−
4
)
,
and
𝑤
⃗
=
(
1
,
2
,
1
)
𝑤⃗= (1,2,1)
w
⃗
=
(
1
,
2
,
1
)
, compute
u
⃗
×
w
⃗
\vec{u}\ \times\ \vec{w}
u
×
w
.
If
u
=
(
2
,
0
,
−
3
)
,
v
=
(
−
2
,
2
,
−
4
)
u=\left(2,0,-3\right),\ v=\left(-2,2,-4\right)
u
=
(
2
,
0
,
−
3
)
,
v
=
(
−
2
,
2
,
−
4
)
and
w
=
(
1
,
2
,
1
)
w=\left(1,2,1\right)
w
=
(
1
,
2
,
1
)
, find a vector that is orthogonal to
u
v
⃗
\vec{uv}
uv
and
u
w
⃗
\vec{uw}
u
w
.
Practice Question: Vector Product
Practice Question: Vector Product
Given that
𝑢
⃗
=
(
1
,
1
,
−
1
)
𝑢⃗=(1,1,−1)
u
⃗
=
(
1
,
1
,
−
1
)
and
𝑣
⃗
=
(
0
,
−
2
,
2
)
,
𝑣⃗=(0,−2,2),
v
⃗
=
(
0
,
−
2
,
2
)
,
compute
(
𝑢
⃗
⋅
𝑣
⃗
)
(
𝑣
⃗
×
𝑢
⃗
)
.
(𝑢⃗\cdot𝑣⃗)(𝑣⃗\times𝑢⃗).
(
u
⃗
⋅
v
⃗
)
(
v
⃗
×
u
⃗
)
.
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.