Learn Vectors with Examples, Equations, and Practice Problems

Let $r (t) = ⟨ 1 + 3 \cos (t), 4 \cos (t), - 1 + 5 \sin (t) ⟩$ . Find an equation for a sphere that this curve lies on.
As it says in the title, any help would be greatly appreciated.

Patricia Bean 2022-07-15

A particle is moving in $R^{3}$ so that its acceleration function is $a (t) = ⟨ 2 t, 1, 0 ⟩$ . Find the position function, r(t) of the particle if it starts at the point $(- 5, 0, 2)$ with initial velocity $v (t) = ⟨ 3, 1, - 1 ⟩$
We have $v (t) = ⟨ 3, 1, - 1 ⟩$ so $r (t) = \int v (t) d t$ = $\int ⟨ 3, 1, - 1 ⟩ d t$ which is $⟨ 3 t, t, - t ⟩$
From here I don't know what to do.

Dawson Downs 2022-07-15

Solve the following linear system in 3-space.
$\begin{aligned} l_{1} & : (x, y, z) = (6, 0, 5) + s (1, 2, 3) \\ l_{2} & : (x, y, z) = (- 1, 7, 14) + t (- 3, 1, 1) \end{aligned}$
My attempt:
First, I converted $l_{1}$ and $l_{2}$ to the parametric form, which is:
$\begin{aligned} l_{1} & : {\begin{cases} x_{1} = 6 + s \\ y_{2} = 2 s \\ z_{1} = 5 + 3 s \end{cases} \\ l_{2} & : {\begin{cases} x_{2} = - 1 - 3 t \\ y_{2} = 7 + t \\ z_{2} = 14 + t \end{cases} \end{aligned}$
Then I set $x_{1} = x_{2}$ and got s+3t=−19.
Then did the same for $y_{1}$ and $y_{2}$ (2s−t=7), and $z_{1}$ and $z_{2}$ (3s−t=9)
From there I got stuck. Any help?

Elsa Brewer 2022-07-14

Why is $κ (A) = \infty$ if A is a singular matrix?
Why the condition number of a singular matrix is ∞. Which also begs the question of what the condition number of a non-invertible non-square matrix would be.

Nelson Jennings 2022-07-14

I have the following vector $z = (\begin{matrix} x \\ y \end{matrix})$ . I also have the function
$f = (\begin{matrix} - 5 β x y \\ 5 β x y \end{matrix})$
I need to rewrite this such that x and y will be simply denoted as the vector z. The closest I am getting to is
$f = 5 β (\begin{matrix} 0 & - x \\ y & 0 \end{matrix}) z$
I can't seem to rearrange this matrix however such that it can be written in terms of z.
I am starting to question whether what I am trying to do is even possible. Is it possible?

Deromediqm 2022-07-14

I have a question on the following:
The line r = i + ( 1 + 3t )j - ( 3 - 4t )k passes through $p_{1} = (1, 1, - 3)$ and is parallel to v = 3j + 4k.
But why can we say that this is parallel? And how can I apply this for different problems.
If someon could help me, it would be very much appreciated

Mbalisikerc 2022-07-14

The following set of points is a lattice in $R^{n}$ :
$D_{n} =$ $(v_{1}, . . ., v_{n}) | v_{1}, . . ., v_{n} \in Z$ and $v_{1} + . . . + v_{n}$ is even}
Find vectors $x_{1}, . . ., x_{k}$ such that $D_{n} = I n t (x_{1}, . . ., x_{k})$
My solution:
Firstly, I know that if $x_{1}, . . ., x_{k} \in R^{n}$ , then we define $I n t (x_{1}, . . ., x_{k}) = t_{1} x_{1} + . . . + t_{k} x_{k}$ where $t_{1}, . . ., t_{k} \in Z$
So, I have deduced that the vectors $x_{1}, . . ., x_{k} \in R^{n}$ so that $D_{n} = I n t (x_{1}, . . ., x_{k})$ are any vectors such that $t_{1} x_{1} + . . . + t_{k} x_{k} = (v_{1}, . . ., v_{n})$ where $v_{1} + . . . + v_{n}$ are even.
However, I feel my solution is certainly lacking and I was wondering if anyone could help guide me in the right direction.

Parker Bird 2022-07-14

Problem looks similar to
$T (x, y, z) = (u, v)$
$u = x + y + z^{2}$
$v = \cos (x) + \tan (y) - z + 3$
vector $r (t) = (t \cdot e, (t^{2} + 2)^{t}, e^{- s^{2}})$
At point t=0 what's the tangent vector of T(r(t))?
Anybody who can help me out with this problem?

rjawbreakerca 2022-07-14

A (2 x 2) matrix $[\begin{matrix} a & c + i d \\ c - i d & b \end{matrix}]$ where a, b, c, d are real constants will have two different eigenvalues unless it is a multiple of the identity matrix.
i used this $| A - λ I | = d e t (A - λ I) = 0$ to prove that it will have 2 eigenvalues. How do i prove that it will only have 1 when it is a multiple of the identity matrix?

gorgeousgen9487 2022-07-13

Given that the $A_{11}$ and $A_{22} \in R^{3 x 3}$ are invertible, $A_{21} \in R^{3 x 3}$ , and $b_{1}, b_{2}, x_{1}, x_{2} \in R^{3}$ , then solve for $x_{1}$ and $x_{2}$ from
$[\begin{matrix} A_{11} & 0 \\ A_{21} & A_{22} \end{matrix}]$ $\begin{aligned} [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] \end{aligned} =$ $\begin{aligned} [\begin{matrix} b_{1} \\ b_{2} \end{matrix}] \end{aligned}$
What are $x_{1}$ and $x_{2}$ in terms of $A_{11}, A_{21}, A_{22}, b_{1}, b_{2}$ ?

Confused Fella2022-05-15

Find the area of the parallelogram with vertices A(0,1), B(3,0), C(5,-2), and D(2,-1).

Lexi Chandler 2022-05-07

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

Let $r (t) = ⟨ 1 + 3 \cos (t), 4 \cos (t), - 1 + 5 \sin (t) ⟩$ . Find an equation for a sphere that this curve lies on.
As it says in the title, any help would be greatly appreciated.

Why is $κ (A) = \infty$ if A is a singular matrix?
Why the condition number of a singular matrix is ∞. Which also begs the question of what the condition number of a non-invertible non-square matrix would be.

I have a question on the following:
The line r = i + ( 1 + 3t )j - ( 3 - 4t )k passes through $p_{1} = (1, 1, - 3)$ and is parallel to v = 3j + 4k.
But why can we say that this is parallel? And how can I apply this for different problems.
If someon could help me, it would be very much appreciated

Problem looks similar to
$T (x, y, z) = (u, v)$
$u = x + y + z^{2}$
$v = \cos (x) + \tan (y) - z + 3$
vector $r (t) = (t \cdot e, (t^{2} + 2)^{t}, e^{- s^{2}})$
At point t=0 what's the tangent vector of T(r(t))?
Anybody who can help me out with this problem?

Find the area of the parallelogram with vertices A(0,1), B(3,0), C(5,-2), and D(2,-1).

(-2,3)+(-1,6)=?

(-4,7)+(4,-4)=?

(5,-7)+(4,5)=?

What is (3,-4) plus (2,-4)?

(-5,4)+(7,7)=?

What is (-7,3) plus (4,4)?

What is (4,-1) plus (4,1)?

(-7,-1)+(-4,-1)=?

(-6,-3)+(-6,-3)=?

Vector Examples, Equations, and Practice Problems