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$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz β}$ |
Related Topics
Wize University Linear Algebra Textbook > Products of Vectors
Dot Product
3 Activities
A vector perpendicular to the vector connecting the points
P
1
(
0
,
1
,
β
2
)
P_1(0,1,-2)
P
1
β
(
0
,
1
,
β
2
)
and
P
2
(
β
2
,
0
,
1
)
P_2(-2,0,1)
P
2
β
(
β
2
,
0
,
1
)
is:
<
β
1
,
β
1
,
β
1
>
\rowvecth{-1}{1}{1}
β¨
β
1
,
1
,
1
β©
<
1
,
β
β
1
,
β
1
>
\rowvecth{1}{-1}{1}
β¨
1
,
β
1
,
1
β©
<
1
,
β
1
,
β
β
1
>
\rowvecth{1}{1}{-1}
β¨
1
,
1
,
β
1
β©
<
1
,
β
1
,
β
1
>
\rowvecth{1}{1}{1}
β¨
1
,
1
,
1
β©
I don't know
Check Submission
More Dot Product Questions:
133 - FML 3 - 18.1W e.g. 20
For two vectors
v
β
\bcb{\vv}
v
and
w
β
\bcb{\vw}
w
β
β β
I
ββ£
R
n
\bcb{\in}\; \R{n}
β
I
R
n
,
v
β
β
w
β
:
=
βΎ
\bcb{\vv \cdot \vw :=\underline{\qquad\qquad\qquad}}
v
β
w
:=
β
.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 40_$\tkcth{Mock F}\tkct{?}$_$\tkct{no soln / vid}$
If
u
β
=
<
1
,
β
2
,
3
>
\bcb{\vec{u} = \left< 1, -2, 3 \right>}
u
=
β¨
1
,
β
2
,
3
β©
and
v
β
=
<
2
,
β
β
1
,
β
3
>
\bcb{\vec{v} = \left< 2,\, -1,\, 3 \right>}
v
=
β¨
2
,
β
1
,
3
β©
, find
(
2
u
β
)
β
(
3
v
β
)
\bcb{(2\vec{u})\cdot (3\vec{v})}
(
2
u
)
β
(
3
v
)
.
Example: Orthogonal Vectors
Example:
Find all values of
k
k
k
such that the vectors
v
β
=
(
k
,
1
)
\vec{v}=\left(k,\ 1\right)
v
=
(
k
,
1
)
and
u
β
=
(
k
,
β
1
)
\vec{u}=\left(k,-1\right)
u
=
(
k
,
β
1
)
are orthogonal.
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find the value(s) of
c
c
c
such that
u
β
=
(
β
1
,
c
,
2
)
\vec{u}=(-1,c,2)\
u
=
(
β
1
,
c
,
2
)
and
v
β
=
(
β
7
,
c
,
β
4
)
\vec{v}=\left(-7,\ c,\ -4\right)
v
=
(
β
7
,
c
,
β
4
)
are orthogonal.
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find the value(s) of
c
c
c
such that
u
β
=
(
β
1
,
c
,
2
)
\vec{u}=(-1,c,2)\
u
=
(
β
1
,
c
,
2
)
and
v
β
=
(
β
7
,
c
,
β
4
)
\vec{v}=\left(-7,\ c,\ -4\right)
v
=
(
β
7
,
c
,
β
4
)
are orthogonal.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 52
If
β£
u
β
β£
=
2
\bcb{|{ \vec{u}| } = 2}
β£
u
β£
=
2
and
β£
v
β
β£
=
3
\bcb{|{ \vec{v} |} = 3}
β£
v
β£
=
3
and
u
β
\bcb{\vec{u}}
u
has the same direction as
v
β
\bcb{\vec{v}}
v
then
(
3
u
β
)
β
(
2
v
β
)
\bcb{(3\vec{u}) \cdot (2\vec{v})}
(
3
u
)
β
(
2
v
)
is:
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | $\tkco{duplicate to mock β}$ 133 - FML 3 - 18.1W e.g. 46_$\tkcth{Mock F}\tkct{?}$_$\tkct{vid soln only}$
If
β£
u
β
β£
=
2
\bcb{|{\vec{u}|} = 2}
β£
u
β£
=
2
and
β£
v
β
β£
=
3
\bcb{|{ \vec{v} |} = 3}
β£
v
β£
=
3
, where
u
β
β
v
β
=
2
\bcb{\vec{u} \cdot \vec{v} = 2}
u
β
v
=
2
, find
β£
u
β
β
v
β
β£
\bcb{|{\vec{u} - \vec{v} }|}
β£
u
β
v
β£
.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 54
Find the value of
k
\bcb{k}
k
such that the vector
u
β
=
<
1
,
k
,
3
>
\bcb{\vec{u} = \left< 1, k, 3 \right>}
u
=
β¨
1
,
k
,
3
β©
is perpendicular to the vector
v
β
=
<
2
,
β
1
,
k
>
\bcb{\vec{v} = \left< 2, -1, k\right>}
v
=
β¨
2
,
β
1
,
k
β©
.
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
β
v
β
\vec{u}\ \bullet\ \vec{v}
u
β
v
.
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
)
.
Find the value(s) of
k
k
k
(if any) such that the vectors
u
β
=
(
1
,
k
,
1
,
0
,
β
1
)
\overrightarrow{u}=\left(1,k,1,0,-1\right)
u
=
(
1
,
k
,
1
,
0
,
β
1
)
and
v
β
=
(
k
,
2
,
0
,
3
,
β
4
)
\overrightarrow{v}=\left(k,2,0,3,-4\right)
v
=
(
k
,
2
,
0
,
3
,
β
4
)
are orthogonal.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz β}$ | Basic dot product.
The dot product
v
β
β
w
β
\vv \cdot \vw
v
β
w
, for
v
β
=
<
β
1
,
β
1
,
β
β
1
>
\vv = \rowvecth{-1}{1}{-1}
v
=
β¨
β
1
,
1
,
β
1
β©
and
w
β
=
<
β
2
,
β
β
2
,
β
3
>
\vw = \rowvecth{-2}{-2}{3}
w
=
β¨
β
2
,
β
2
,
3
β©
, is equal to:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz β}$ |
A unit vector perpendicular to the vector
w
β
=
<
2
,
β
5
>
\vw = \rowvec{2}{5}
w
=
β¨
2
,
5
β©
is:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz β}$ |
A vector perpendicular to the vector connecting the points
P
1
(
2
,
2
,
2
)
P_1(2,2,2)
P
1
β
(
2
,
2
,
2
)
and
P
2
(
1
,
β
1
,
1
)
P_2(1,-1,1)
P
2
β
(
1
,
β
1
,
1
)
is:
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.
Unit Vectors: Dot Product
Consider the vectors
u
=
i
β
4
k
\bm{u}=\bm{i}-4\bm{k}
u
=
i
β
4
k
and
v
=
3
i
β
2
j
+
k
\bm{v}=3\bm{i}-2\bm{j}+\bm{k}
v
=
3
i
β
2
j
+
k
in
R
3
\reals^3
R
3
. Find (if possible)
3
v
β
(
k
β
2
u
)
3\bm{v}\cdot(\bm{k}-2\bm{u})
3
v
β
(
k
β
2
u
)
.