If $A=-{a}_{x}+6{a}_{y}+5{a}_{z}andB={a}_{x}+2{a}_{y}+3{a}_{x}$, find (a) the scalar projection of A on B

Ismael Molina
2022-07-25
Answered

If $A=-{a}_{x}+6{a}_{y}+5{a}_{z}andB={a}_{x}+2{a}_{y}+3{a}_{x}$, find (a) the scalar projection of A on B

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asked 2021-02-11

Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in $M}_{2\times 4$ with the property that FA = 0 (the zero matrix in ${M}_{3\times 4})$ . Determine if H is a subspace of $M}_{2\times 4$

asked 2021-05-17

Find the scalar and vector projections of b onto a.

$a=(4,7,-4),b=(3,-1,1)$

asked 2021-05-29

Which of the following expressions are meaningful? Which are meaningless? Explain.

a)$(a\cdot b)\cdot c$

$(a\cdot b)\cdot c$ has ? because it is the dot product of ?.

b)$(a\cdot b)c$

$(a\cdot b)c$ has ? because it is a scalar multiple of ?.

c)$|a|(b\cdot c)$

$|a|(b\cdot c)$ has ? because it is the product of ?.

d)$a\cdot (b+c)$

$a\cdot (b+c)$ has ? because it is the dot product of ?.

e)$a\cdot b+c$

$a\cdot b+c$ has ? because it is the sum of ?.

f)$|a|\cdot (b+c)$

$|a|\cdot (b+c)$ has ? because it is the dot product of ?.

a)

b)

c)

d)

e)

f)

asked 2021-05-29

Find a vector equation and parametric equations for the line segment that joins P to Q.

P(0, - 1, 1), Q(1/2, 1/3, 1/4)

P(0, - 1, 1), Q(1/2, 1/3, 1/4)

asked 2022-07-20

Show that vectors are in an affine plane if and only if det=0

Let $\mathbb{K}$ be a field and, $a=\left(\begin{array}{l}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right),b=\left(\begin{array}{l}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}\right),c=\left(\begin{array}{c}{c}_{1}\\ {c}_{2}\\ {c}_{3}\end{array}\right),d=\left(\begin{array}{c}{d}_{1}\\ {d}_{2}\\ {d}_{3}\end{array}\right)\in {\mathbb{K}}^{3}$

Show that a,b,c,d are in an affine plane if and only if

$det\left(\begin{array}{llll}{a}_{1}& {b}_{1}& {c}_{1}& {d}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}& {d}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}& {d}_{3}\\ 1& 1& 1& 1\end{array}\right)=0$

How can I show this?

Let $\mathbb{K}$ be a field and, $a=\left(\begin{array}{l}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right),b=\left(\begin{array}{l}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}\right),c=\left(\begin{array}{c}{c}_{1}\\ {c}_{2}\\ {c}_{3}\end{array}\right),d=\left(\begin{array}{c}{d}_{1}\\ {d}_{2}\\ {d}_{3}\end{array}\right)\in {\mathbb{K}}^{3}$

Show that a,b,c,d are in an affine plane if and only if

$det\left(\begin{array}{llll}{a}_{1}& {b}_{1}& {c}_{1}& {d}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}& {d}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}& {d}_{3}\\ 1& 1& 1& 1\end{array}\right)=0$

How can I show this?

asked 2022-08-22

The problem defines:

velocity in the local base $\{{e}_{r},{e}_{\theta},{e}_{z}\}$, the gradient and acceleration in the local curvilignear base ${e}_{i}\otimes {e}_{j}$ avec $i,j\in \{r,\theta ,z\}$

In a problem's solution, this result is used to calculate the stress resultant

$\sigma \overrightarrow{n}=\left[\begin{array}{l}0\\ \mu w/H\\ -{p}_{atm}\end{array}\right]$

$R={\int}_{0}^{2\pi}$

$+{\int}_{0}^{2\pi}{\int}_{0}^{{R}_{d}}{p}_{atm}\cdot {\overrightarrow{e}}_{z}rdrd\theta =$

$={\int}_{0}^{2\pi}({\overrightarrow{e}}_{\theta}(\theta )d\theta {\int}_{0}^{{R}_{d}}\frac{\mu \omega}{H}rdr-\left(2\pi {p}_{\text{atm}}{\int}_{0}^{{R}_{d}}rdr\right){\overrightarrow{e}}_{z}$

then it says ${\int}_{0}^{2\pi}({\overrightarrow{e}}_{\theta}(\theta )d\theta =0$ which I've been stuck for one hour looking where it came from ?

velocity in the local base $\{{e}_{r},{e}_{\theta},{e}_{z}\}$, the gradient and acceleration in the local curvilignear base ${e}_{i}\otimes {e}_{j}$ avec $i,j\in \{r,\theta ,z\}$

In a problem's solution, this result is used to calculate the stress resultant

$\sigma \overrightarrow{n}=\left[\begin{array}{l}0\\ \mu w/H\\ -{p}_{atm}\end{array}\right]$

$R={\int}_{0}^{2\pi}$

$+{\int}_{0}^{2\pi}{\int}_{0}^{{R}_{d}}{p}_{atm}\cdot {\overrightarrow{e}}_{z}rdrd\theta =$

$={\int}_{0}^{2\pi}({\overrightarrow{e}}_{\theta}(\theta )d\theta {\int}_{0}^{{R}_{d}}\frac{\mu \omega}{H}rdr-\left(2\pi {p}_{\text{atm}}{\int}_{0}^{{R}_{d}}rdr\right){\overrightarrow{e}}_{z}$

then it says ${\int}_{0}^{2\pi}({\overrightarrow{e}}_{\theta}(\theta )d\theta =0$ which I've been stuck for one hour looking where it came from ?

asked 2022-09-22

Given a vector $v\in {\mathbb{R}}^{n}$ then show that we can express $\sum _{k}{\omega}_{k}{v}_{k}^{2}$ as a matrix product of the form ${v}^{T}Mv$. Give an expression for M in terms of $\omega =[{\omega}_{1}...{\omega}_{n}{]}^{T}$

I understand that here, the product of a row vector, matrix, and column vector (in that order) is a scalar. However, how do I write M in terms of $\omega =[{\omega}_{1}...{\omega}_{n}{]}^{T}$?

I understand that here, the product of a row vector, matrix, and column vector (in that order) is a scalar. However, how do I write M in terms of $\omega =[{\omega}_{1}...{\omega}_{n}{]}^{T}$?