Learn Vectors with Examples, Equations, and Practice Problems

Here's the question: Let Q be a square $n \times n$ matrix. Let ${e_{1}, e_{2}, . . ., e_{n}}$ be the n standard basis column vectors of Rn. Show that the set of vectors ${Q e_{1}, Q e_{2}, . . ., Q e_{n}}$ also form a set of orthonormal vectors.
In terms of my attempts, I've proven that each column vector of Q forms a set of orthonormal vectors in $R^{n}$ . I feel like this may be very close but I'm struggling to picture where to go from here. If this method is correct, where do I go from here? If this method is not correct, what would be the best way to prove this?

corniness9a 2022-09-06

The vector $u_{λ}$ is assumed to be normalized.
Then I want to understand How these two conditions are satisfied?
$\begin{aligned} \sum_{k} u_{k κ}^{} u_{k λ} = δ_{κ λ} \\ \sum_{κ} u_{k κ}^{} u_{l κ} = δ_{k l} \end{aligned}$

Taylor Church 2022-09-06

$Δ (x \cdot \nabla u) = Δ u + x \cdot (\nabla (Δ u))$
Does this hold for $x \in R^{n}$ and $u : R^{n} \to R$ ? If it does, why does it?
I calculated the above formula by writing out all the variables from $x_{1}$ to $x_{2}$ and guessing that it must be correct. But I am not familiar with 'direct' calculations using $\nabla$ and $Δ$ , so I need some help.

Genesis Gibbs 2022-09-06

I am trying to prove that $A A^{+} {\vec{a}}_{i}^{} = {\vec{a}}_{i}^{}$ , $\forall i = 1.. n$ is equal to the more popular version $A A^{+} A = A$ . I started by setting up a system of equations as follows:
${\begin{cases} A A^{+} {\vec{a}}_{1}^{} = {\vec{a}}_{1}^{} \\ ⋮ \\ A A^{+} {\vec{a}}_{n}^{} = {\vec{a}}_{n}^{} \end{cases}$
Then I turned it into a matrix form:
$[\begin{matrix} A A^{+} {\vec{a}}_{1}^{} \\ ⋮ \\ A A^{+} {\vec{a}}_{n}^{} \end{matrix}] = [\begin{matrix} {\vec{a}}_{1}^{} \\ ⋮ \\ {\vec{a}}_{n}^{} \end{matrix}]$
$A A^{+} [\begin{matrix} {\vec{a}}_{1}^{} \\ ⋮ \\ {\vec{a}}_{n}^{} \end{matrix}] = [\begin{matrix} {\vec{a}}_{1}^{} \\ ⋮ \\ {\vec{a}}_{n}^{} \end{matrix}]$
Since ${\vec{a}}_{i}^{}$ is a column vector it has shape (m x 1) which seems to produce weird vector in the last equation. Is this part of a "proof" any good or is it a bad idea from the begining?

Leonel Schwartz 2022-09-06

What is the explicit expression of a plane wave in the frequency domain?
A plane wave in the time domain can be written (using notation for an electric field):
$E (r, t) = E_{0} e^{i (k r - ω t)}$
what is the corresponding expression for a plane wave, $E (r, ω)$ , in the frequency domain?

s2vunov 2022-09-05

Consider the vectors $\vec{u}$ = 2 $\vec{i}$ + $\vec{j}$ + $\vec{k}$ and $\vec{v}$ = $\vec{i}$ +2 $\vec{j}$ .
a) Determine a positive orthornomal basis { $\vec{a}$ , $\vec{b}$ , $\vec{c}$ } with $\vec{a}$ parallel to $\vec{u}$ and $\vec{b}$ coplanar with $\vec{u}$ and $\vec{v}$ .
b) Determine the coordinates of $\vec{w}$ = 3 $\vec{i}$ +4 $\vec{j}$ +5 $\vec{k}$ in the orthonormal basis { $\vec{a}$ , $\vec{b}$ , $\vec{c}$ }.
I'm stuck in how i would find $\vec{b}$ and $\vec{c}$ .

Tiana Hill 2022-09-05

Let $v_{1} = (\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix})$ , $v_{2} = (\begin{matrix} 0 \\ 1 \\ 1 \\ 0 \end{matrix})$ , $v_{3} = (\begin{matrix} 0 \\ 0 \\ 1 \\ 1 \end{matrix})$ , $v_{4} = (\begin{matrix} 2 \\ 0 \\ 0 \\ 1 \end{matrix})$ $R^{4}$ vectors.
Show that every $v \in R^{4 \times 1}$ can be written as vectors $(v_{1}, v_{2}, v_{3}, v_{4})$ linear combination.
My attempt:
$[\begin{array}{ccccl} 1 & 0 & 0 & 2 & v_{1} \\ 1 & 1 & 0 & 0 & v_{2} \\ 0 & 1 & 1 & 0 & v_{3} \\ 0 & 0 & 1 & 1 & v_{4} \end{array}]$
Where do I go from here? Every input is appreciated.

Drew Williamson 2022-09-04

5) Write the equation of the line through the points P(1,3,7) and Q(2,5,11) in: a) Vector form b) Parametric form c) Symmetric form

dripcima24 2022-09-03

If I want to define the line y=2x when discussing 3D lines this is actually a plane. We should be able to turn it into a line with cartesian equation x=2y=z/0 but that would give us something undefined. How do you resolve that problem?

Aidyn Crosby 2022-09-01

Given a set of sets of vectors $V = {V_{1} = {v_{11}, v_{12}, . ., v_{1 n}}, . . ., V_{m} {v_{m 1}, v_{m 2}, . ., v_{m n}}}$ , $v_{i j} \in {- 1, 0, 1}^{r}$ . We choose one vector from each set, so that the maximum component of their sum is minimal. Can you advise on how best to approximate the optimal solution of the problem?

Corinne Woods 2022-09-01

Consider the point $P (x_{0}, y_{0}, z_{0})$
We have already found the equation of a line that contains mypoint and the origin. We have also found an equation of a line thatcontains my point and the point (1, -1, 1).
Find an equation of the plane that contains the two lines you have just found.

depistetzwy 2022-08-31

Given a vector $u = (x, y, z)$ what is $‖ u ‖_{2}^{2}$ ?

embelurildmixjm 2022-08-30

If a and b are vectors perpendicular to each other Then is it okay to say that the general solution to $\vec{r} \times \vec{a} = \vec{b}$ is
$\vec{r} = \frac{\vec{r} \cdot \vec{a}}{\vec{a} \cdot \vec{a}} \vec{a} + \frac{1}{\vec{a} \cdot \vec{a}} (\vec{a} \times \vec{b}) ?$
I did cross product with a vector to the given vector equation to get the above result will we really say the solution is that only as such x has dependance on r again isn't ?

sponsorjewk 2022-08-29

If u=(2,-6,k) and v=(-1,3,2) are parallel, what is k?

zibazeleor 2022-08-29

I am doing some review on cross products, and I forgot how do a cross product similar to this:
$v Q \times B_{1} = v Q \times B_{2}$
where $Q, B_{1}, B_{2}$ are vectors, $\times$ is cross product, and v is a scalar.
I would be trying to prove that $B_{1}$ and $B_{2}$ are equal to each other, other than saying that it is indeed true, but I don't know if an "inverse" cross product exists, and can't seem to figure out how to prove it exists, if it does. Any help would be awesome!

Expositur3e 2022-08-29

Finding $\ker [A]^{T}$
Let
$A = (\begin{matrix} 1 & 1 & - 1 & 2 \\ 1 & 1 & 0 & 3 \\ - 1 & 0 & 1 & 0 \end{matrix}) .$
I have to calculate $\ker [A]$ and $\ker [A]^{T}$
I proved that $\dim (I m [A]) = 3$ with basis
$[\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}]$
and $\dim (\ker [A]) = 1$ with basis
$[\begin{matrix} - 1 \\ - 2 \\ - 1 \\ 1 \end{matrix}],$
but i have problems with $\ker [A]^{T}$ . Do i have to calculate $A^{T}$ and then, with that matrix, finding the ker?

Nina Perkins 2022-08-27

The line
$\frac{x - 2}{2} = \frac{y + 1}{- 2} = \frac{z + 1}{- 1}$
Is the axis of a circular cone with vertex on the xy-plane. Find the equation of the cone, if the point $M_{1} (1, 1, - \frac{5}{2})$ is on the surface.
Thus far, I have found that the vertex is the point V(0,1,0), and that the radius of the cone at the point $M_{1}$ is $\sqrt{5}$

ngombangouh 2022-08-26

So it's easy to take the tensor product of two vectors,
$[\begin{matrix} a \\ b \end{matrix}] \otimes [\begin{matrix} c \\ d \end{matrix}] = [\begin{matrix} a c \\ a d \\ b c \\ b d \end{matrix}]$
but it seems much more difficult to go backwards, to "factor" the vector. Is there a method to find two vectors whose tensor product is a given vector?
In my specific problem, the magnitude of the resultant vector is 1, if that's relevant.

Miguel Mathis 2022-08-25

1.Given a circle C with center A and radius r.
2.Given a line D with a vector u passing through point $P_{0}$ .
3.Knowing that P is on D only if $P = P_{0} + t u$
4.Knowing that P is on C if $‖ {P - A}^{2} ‖ = r^{2}$
Prove that point P is on C and D if there exists a real number t where
$[‖ u ‖^{2}] t^{2} + [2 (P_{0} - A) \cdot u] t + [‖ {P_{0} - A}^{2} ‖ - r^{2}] = 0.$
What properties should I be using in order to solve this?

rubinowyac 2022-08-24

Resolution of $2 \hat{i} + 3 \hat{j}$ along $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$
I proceeded as below:
Let A, a and b be the given vectors in the same order. Let
$A = λ a + μ b$
Putting the values and rearranging,
$2 \hat{i} + 3 \hat{j} = (λ + μ) \hat{i} + (λ - μ) \hat{j}$
Then, noting that two vectors are equal iff their magnitudes as well as directions are equal, first we equate the directions and taking tan of both sides.
$\frac{3}{2} = \frac{λ - μ}{λ + μ}$
Thus, $λ = - 5 μ$
Now, taking magnitudes and squaring,
$13 = 2 (λ^{2} + μ^{2})$
Using our previous equation and doing some work, we get
$λ = \pm \frac{5}{\sqrt{2}} and μ = \mp \frac{1}{\sqrt{2}}$

Vectors in the Precalculus course are usually more challenging since there are different vectors examples that are always mentioned. For example, if you are majoring in Engineering disciplines, you will have to use more than one approach to explain the most efficient ways. Take a look at vectors practice problems that have been presented below. It will help you learn and find the answers that will let you see the best equation and graphs. At the same time, when you are dealing with vectors equations, do not forget about unknown coefficients as you are looking for combinations and various solutions.

Precalculus

Vector Examples, Equations, and Practice Problems