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Consider three vectors A=2i+j,,,B=i-2j, and C=-2i+j. The angle between C and ABβ¦
Related Topics
Wize University Linear Algebra Textbook > Products of Vectors
Angle Between Vectors
6 Activities
Wize University Linear Algebra Textbook > Products of Vectors
Cross Product
3 Activities
Consider three vectors
A
β
=
2
i
^
+
j
^
,
β
β
B
β
=
i
^
β
2
j
^
\vec{A}=2\hat{i}+\hat{j},\,\,\vec{B}=\hat{i}-2\hat{j}
A
=
2
i
^
+
j
^
β
,
B
=
i
^
β
2
j
^
β
, and
C
β
=
β
2
i
^
+
j
^
\vec{C}=-2\hat{i}+\hat{j}
C
=
β
2
i
^
+
j
^
β
. The angle between
C
β
\vec{C}
C
and
A
β
Γ
B
β
\vec{A}\times\vec{B}
A
Γ
B
is?
45
Β°
45\degree
45Β°
0
Β°
0\degree
0Β°
180
Β°
180\degree
180Β°
90
Β°
90\degree
90Β°
I don't know
Check Submission
More Angle Between Vectors Questions:
133 - FML 3 - 18.1W e.g. 47
Given
u
β
=
<
1
,
0
,
1
>
\bcb{\vec{u} = \left< 1, 0, 1\right>}
u
=
β¨
1
,
0
,
1
β©
and
v
β
=
<
1
,
0
,
0
>
\bcb{\vec{v} = \left< 1, 0, 0\right>}
v
=
β¨
1
,
0
,
0
β©
, compute the acute angle between
u
β
\bcb{\vec{u}}
u
and
v
β
\bcb{\vec{v}}
v
.
Find the value of cos π where π is the acute angle between
π’
β
=
(
1
,
β
2
)
π’β=(1,β2)
u
β
=
(
1
,
β
2
)
and
π£
β
=
(
3
,
1
)
.
\vecπ£ = (3,1).
v
=
(
3
,
1
)
.
Additional practice problems--vector products
Additional Practice Problems--Vector Products
1. If
u
β
=
(
β
2
,
3
,
β
1
)
,
v
β
=
(
β
1
,
β
2
,
3
)
\vec{u}=\left(-2,3,-1\right),\ \ \vec{v}=\left(-1,-2,3\right)
u
=
(
β
2
,
3
,
β
1
)
,
v
=
(
β
1
,
β
2
,
3
)
, find
a)
u
β
β
v
β
\vec{u}\ \bullet\ \vec{v}
u
β
v
More Cross Product Questions:
Cross Product
Practice: Orthogonal Vectors
Find a vector orthogonal to both
u
β
=
[
2
β
1
3
]
\vec u = \begin{bmatrix} 2\\ -1\\ 3 \end{bmatrix}
u
=
β
2
β
1
3
β
β
and
v
β
=
[
0
β
1
β
1
]
\vec{v} = \begin{bmatrix} 0\\ -1\\ -1\\ \end{bmatrix}
v
=
β
0
β
1
β
1
β
β
.
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
,
β
3
)
\left(1,2,-3\right)
(
1
,
2
,
β
3
)
and is orthogonal to the lines with equations:
(
x
,
y
,
z
)
=
(
β
5
,
0
,
1
)
+
t
(
1
,
0
,
β
1
)
(x,y,z)= (-5,0,1)+t(1,0,-1)
(
x
,
y
,
z
)
=
(
β
5
,
0
,
1
)
+
t
(
1
,
0
,
β
1
)
and
(
x
,
y
,
z
)
=
(
3
,
β
1
,
6
)
+
t
(
0
,
2
,
1
)
(x,y,z)=(3,-1,6)+t(0,2,1)
(
x
,
y
,
z
)
=
(
3
,
β
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
[
β
3
,
1
,
0
]
[-3,1,0]
[
β
3
,
1
,
0
]
.
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find a vector orthogonal to both
π’
β
=
(
2
,
β
1
,
3
)
π’β=(2,β1,3)
u
β
=
(
2
,
β
1
,
3
)
and
π£
β
=
(
0
,
β
1
,
β
1
)
.
\vec{π£} = (0, β1, β1).
v
=
(
0
,
β
1
,
β
1
)
.
Evaluate
2
k
^
Γ
(
3
i
^
Γ
2
j
^
)
2\hat{k}\times(3\hat{i}\times2\hat{j})
2
k
^
Γ
(
3
i
^
Γ
2
j
^
β
)
Practice: Cross Product
Practice: Cross Product
Recall that
i
β
=
[
1
,
0
,
0
]
\vec{i}=\left[1,\ 0,\ 0\right]
i
=
[
1
,
0
,
0
]
,
j
β
=
[
0
,
1
,
0
]
\vec{j}=\left[0,\ 1,\ 0\right]
j
β
=
[
0
,
1
,
0
]
, and
k
β
=
[
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
β
=
[
2
β
1
3
]
\vec u = \begin{bmatrix} 2\\ -1\\ 3 \end{bmatrix}
u
=
β
2
β
1
3
β
β
and
v
β
=
[
0
β
1
β
1
]
\vec{v} = \begin{bmatrix} 0\\ -1\\ -1\\ \end{bmatrix}
v
=
β
0
β
1
β
1
β
β
.
For extra practice, check your answer!
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find a vector orthogonal to both
π’
β
=
(
2
,
β
1
,
3
)
π’β=(2,β1,3)
u
β
=
(
2
,
β
1
,
3
)
and
π£
β
=
(
0
,
β
1
,
β
1
)
.
\vec{π£} = (0, β1, β1).
v
=
(
0
,
β
1
,
β
1
)
.
Practice: Orthogonal Vectors
Practice: Orthogonal Vectors
Find a vector orthogonal to both
π’
β
=
(
2
,
β
1
,
3
)
π’β=(2,β1,3)
u
β
=
(
2
,
β
1
,
3
)
and
π£
β
=
(
0
,
β
1
,
β
1
)
.
\vec{π£} = (0, β1, β1).
v
=
(
0
,
β
1
,
β
1
)
.
Practice: Cross Product
Practice: Cross Product
Consider
u
β
=
[
β
2
,
3
,
β
1
]
\vec{u}=\left[-2,\ 3,\ -1\right]
u
=
[
β
2
,
3
,
β
1
]
and
v
β
=
[
1
,
0
,
β
1
]
\vec{v}=\left[1,\ 0,\ -1\right]
v
=
[
1
,
0
,
β
1
]
.
If
u
β
=
(
2
,
0
,
β
3
)
,
v
β
=
(
β
2
,
2
,
β
4
)
\vec{u}=\left(2,0,-3\right),\ \vec{v}=\left(-2,2,-4\right)
u
=
(
2
,
0
,
β
3
)
,
v
=
(
β
2
,
2
,
β
4
)
and
w
β
=
(
1
,
2
,
1
)
\vec{w}=\left(1,2,1\right)
w
=
(
1
,
2
,
1
)
, find a vector that is orthogonal to
u
v
β
\vec{uv}
uv
and
u
w
β
\vec{uw}
u
w
.
If
π’
β
=
(
2
,
0
,
β
3
)
,
π£
β
=
(
β
2
,
2
,
β
4
)
,
π’β = (2, 0, β3), \vecπ£ = (β2, 2, β4),
u
β
=
(
2
,
0
,
β
3
)
,
v
=
(
β
2
,
2
,
β
4
)
,
and
π€
β
=
(
1
,
2
,
1
)
π€β= (1,2,1)
w
β
=
(
1
,
2
,
1
)
, compute
u
β
Γ
w
β
\vec{u}\ \times\ \vec{w}
u
Γ
w
.
Give that
π’
β
π’β
u
β
and
π£
β
\vecπ£
v
are vectors in
R
3
\mathbb{R}^3
R
3
, which of the following statements is/are always true?
i.
2
π’
β
+
3
π£
β
=
2
π£
β
+
3
π’
β
2π’β + 3\vecπ£ = 2\vecπ£ + 3π’β
2
u
β
+
3
v
=
2
v
+
3
u
β
ii.
2
π’
β
.
(
β
3
π£
β
)
=
β
2
(
3
π’
β
.
π£
β
)
2π’β . (β3\vecπ£ ) = β2(3π’β . \vecπ£ )
2
u
β.
(
β
3
v
)
=
β
2
(
3
u
β.
v
)
If
π’
β
=
(
2
,
0
,
β
3
)
,
π£
β
=
(
β
2
,
2
,
β
4
)
,
π’β = (2, 0, β3), \vecπ£ = (β2, 2, β4),
u
β
=
(
2
,
0
,
β
3
)
,
v
=
(
β
2
,
2
,
β
4
)
,
and
π€
β
=
(
1
,
2
,
1
)
π€β= (1,2,1)
w
β
=
(
1
,
2
,
1
)
, compute
u
β
Γ
w
β
\vec{u}\ \times\ \vec{w}
u
Γ
w
.
If
u
=
(
2
,
0
,
β
3
)
,
v
=
(
β
2
,
2
,
β
4
)
u=\left(2,0,-3\right),\ v=\left(-2,2,-4\right)
u
=
(
2
,
0
,
β
3
)
,
v
=
(
β
2
,
2
,
β
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
β
\vec{uv}
uv
and
u
w
β
\vec{uw}
u
w
.
Practice Question: Vector Product
Practice Question: Vector Product
Given that
π’
β
=
(
1
,
1
,
β
1
)
π’β=(1,1,β1)
u
β
=
(
1
,
1
,
β
1
)
and
π£
β
=
(
0
,
β
2
,
2
)
,
π£β=(0,β2,2),
v
β
=
(
0
,
β
2
,
2
)
,
compute
(
π’
β
β
π£
β
)
(
π£
β
Γ
π’
β
)
.
(π’β\cdotπ£β)(π£β\timesπ’β).
(
u
β
β
v
β
)
(
v
β
Γ
u
β
)
.
Dot and Cross Products: Vector Norm
If
u
β
,
v
β
,
w
β
β
R
3
\vec{u},\ \vec{v},\ \vec{w}\ \ \in\ R^3
u
,
v
,
w
β
R
3
, which of the following operations is/are defined?
(i.e. which of the following can be calculated?)
Select all that are correct.