Learn Vectors with Examples, Equations, and Practice Problems

Geometrical Interpretation of Vectors Addition
I don't understand what it means by $(y_{1} - x_{1}, y_{2} - x_{2}, y_{3} - x_{3})$
This is the part of the book:
The directed line segment PQ, from the point $P = (x_{1}, x_{2}, x_{3})$ to the point $Q = (y_{l}, y_{2}, y_{3})$ , has the same length and direction as the directed line segment from the origin $0 = (0, 0, 0)$ to the point $(y_{l} - x_{1}, y_{2} - x_{2}, y_{3} - x_{3})$ . Furthermore, this is the only segment emanating from the origin which has the same length and direction as PQ. Thus, if one agrees to treat only vectors which emanate from the origin, there is exactly one vector associated with each given length and direction.

Bruce Sherman 2022-10-09

Another proof of $| \vec{a} \times \vec{b} |^{2} = | \vec{a} |^{2} \cdot | \vec{b} |^{2} - (\vec{a} \cdot \vec{b})^{2}$
I know how to prove the formula $| \vec{a} \times \vec{b} |^{2} = | \vec{a} |^{2} \cdot | \vec{b} |^{2} - (\vec{a} \cdot \vec{b})^{2}$ ,but i need to prove it using the formula $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a}$ , how to prove it?

hexaedru8p 2022-10-09

If a and c are unit vectors at an angle $π / 3$ with each other and $(a \times (b \times c)) . (a \times c) = 5$ , then what is the value of [a b c]? I just know the basic meaning of what are vector and scalar triple product. How do I do this question?

hazbijav6 2022-10-09

I've been working on computational fluid dynamics and have come across the following term in index notation:
$\frac{\partial u_{i}}{\partial x_{m}} \frac{\partial u_{j}}{\partial x_{m}}$
However, I'm having a hard time finding a vector notation equivalent to this operation. This is definitely not the inner product or outer product, but kind of like a "right" inner product. Has anyone come across any term like this and its vector notation equivalent?
To be more precise, I would like to know the operation $⊙$ in:
$(\nabla \otimes u) ⊙ (\nabla \otimes u)$
If one exists, or some other form. In the above, $\nabla \equiv \partial / \partial x_{m}$ , u is a Cartesian 3-vector, and $\otimes$ is the direct product.

charlygyloavao9 2022-10-08

let r,c> some real numbers, and $X (t) = ⟨ r \cos t, r \sin t, c t ⟩$ is a curve.
Find the curvature as a function of t
I think that the curvature of a curve is
$‖ \frac{d T}{d s} ‖ = \frac{‖ T^{'} (t) ‖}{‖ X^{'} (t) ‖} Where T = \frac{X^{'} (t)}{‖ X^{'} (t) ‖}$
Since
$X^{'} (t) = ⟨ - r \sin t, r \cos t, c ⟩$
and
$T (t) = \frac{1}{\sqrt{r^{2} + c^{2}}} ⟨ - r \sin t, r \cos t, c ⟩$
It follows that
$T^{'} (t) = \frac{1}{\sqrt{r^{2} + c^{2}}} ⟨ - r \cos t, - r \sin t, 0 ⟩$
Hence
$‖ \frac{d T}{d s} ‖ = \frac{r}{\sqrt{r^{2} + c^{2}}} \frac{1}{\sqrt{r^{2} + c^{2}}} = \frac{r}{r^{2} + c^{2}}$
But apparently, this is not a function of t, Does this mean that the curvature is constant?

Gunsaz 2022-10-08

Write the set of vectors that are orthogonal to v as a linear combination of two unit vectors.
$v = ⟨ 1, - \sqrt{8}, - \sqrt{8} ⟩ is a vector.$ is a vector.
I know I have to find two unit vectors u and w so that any vector that is orthogonal to v can be expressed as a linear combination of u and w.
Help, please.

charlygyloavao9 2022-10-08

Find the direction cosines of the a vector in the plane of $\vec{b} =< 2, 1, 3 >$ and $\vec{c} =< - 1, 2, 1 >$ and perpendicular to $\vec{c}$
Let the vector $\vec{a} = ⟨ x, y, z ⟩$
Since it is perpendicular to $\vec{c}$
$- x + 2 y + z = 0$
And
$| \begin{matrix} x & y & z \\ 2 & 1 & 3 \\ - 1 & 2 & 1 \end{matrix} |$
$⟹ - 5 x - 7 y + 5 z = 0$
since a, b, c are coplanar
So y=0
Then x=y
So the answer should be $⟨ \frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}} ⟩$ , but that is not correct.
What is wrong with my solution?

dekkyrkex4 2022-10-08

Let x be a block vector with two vector elements x=(a,b), where a and b are vectors of size n and m, respectively.
How do I express $r m s (x)$ in terms of $r m s (a)$ , $r m s (b)$ , $m$ and $n$ ?

hazbijav6 2022-10-08

If $α_{i j} A^{i} B^{j} = 0$ and $A^{i}$ and $B^{j}$ are arbitrary vectors, then prove that $α_{i j} = 0$
This problem appeared in my Differential Geometry class, the professor explained the problem by, first taking an arbitrary vector and demonstrating that $α_{i i} = 0$ . and then proceeded to demonstrate that, $A^{l} = B^{m} = 1, (1 ≦ l \leq n, 1 ≦ m ≦ n, l \neq m)$ . I get the proof somewhat. Can any of you elucidate it or give an alternative proof?

revitprojectb7 2022-10-08

If A and B are vectors in $C^{n}$ , prove that $- 2 \leq \frac{A \cdot B + \bar{A \cdot B}}{| A | | B |} \leq 2$

dannyboi2006tk 2022-10-08

Let L be the line of intersection between the planes x + 3y = 7 and 2y + z = 4, and let P be the point (3, 1, 1).
Determine the equation (in normal form) of the plane containing L and P.
I struggle with getting vectors from the two planes to multiply, can anyone show me how?
$L = (1, 2, 0) + t (3, - 1, 2)$ . But i somehow fail to see how my book got that.

Jannek93 2022-10-08

Is there any relation between Gram-Schmidt process in $R^{3}$ and vector cross product?
Using Gram-Schmidt orthogonalization process we can find an orthogonal set of vectors from a given set of vectors,also we were taught previously that crossing between two non-collinear vectors gives a vector perpendicular to the two vectors.Is there any correlation between the two processes of find orthogonal system of vectors,are the two essentially the same?

Jamarcus Lindsey 2022-10-08

I came across a problem while proving a very simple statement.
We have three vectors $T = {v_{1}, v_{2}, v_{3}}$ , and the set of their sums $S = {v_{1} + v_{2}, v_{1} + v_{3}, v_{2} + v_{3}}$ for $v_{i} \in V$ , for V any vector space. I have to prove that for rational coefficients $a, b, c \in Q$ , the following statement is true:
T linearly independent $\Leftrightarrow$ S linearly independent
$\Rightarrow$ was quite easy and I had no problem calculating it.
However, $\Leftarrow$ is where I faced the problem and I would appreciate any help and besides that, does $\Leftarrow$ is true for real coefficients.

fofopausiomiava 2022-10-07

I am trying to program this but I need to verify if the equations I have are correct
$f (x) = | | x | |^{3} + \frac{m}{2} \times | | x | |^{2}$
$gradient of f(x) = 3 | | x | |^{2} + m \times x$
$hessian= 6 | | x | | + m$
if this understanding is correct.
And I start with x in $R^{5}$ i.e. a $5 \times 1$ vector, when I solve for hessian using equation 3, I don't get a matrix. I only get a row vector ( $1 \times 5$ )
||x|| implies Euclidean norm, m is a scalar (but I have been making m a $1 \times 5$ vector to match dimensions of x in my code)

ivybeibeidn 2022-10-07

Let $‖ \cdot ‖ : R^{n} \to R$ be the Euclidean norm. Prove that for any two vectors $x, y \in R^{n}$ with $x \neq 0 \neq y$ the following inequality holds:
$| | \frac{x}{‖ x ‖} - \frac{y}{‖ y ‖} | | \leq \frac{2}{‖ x ‖ + ‖ y ‖} ‖ x - y ‖ .$
I'm very curious as to how I can attack this question. Any hints as well as a full answer is appreciated. I've tried using the triangle inequality (which is, basically, all I could think about) with no luck. I guess I could try using the definition of the euclidean norm letting $x = (x_{1}, \dots, x_{n})$ and $y = (y_{1}, \dots, y_{n})$ and try something, but I don't want to go down that path (and I don't think it would lead me anywhere, either).

Bridger Holden 2022-10-06

Solving and verification of : $\frac{\partial}{\partial y} \cos (x^{T} y) x^{T}$
I am trying to simplify an expression and would like to ask if my approach is correct and also could I simplify this expression further. If so what would I use? I am doing a bit of self-studying here so I must apologize if this is trivial.
Question:
Say $x \in R^{n}$ and $y \in R^{n}$ find the explicit expression:
$\begin{aligned} \frac{\partial}{\partial y} \cos (x^{T} y) x^{T} \end{aligned}$
My Answer:
$\begin{aligned} \frac{\partial}{\partial y} \cos (x^{T} y) x^{T} = - y \sin (x^{T} y) x^{T} \end{aligned}$
Is the answer correct?
Can I simplify this further?

hexaedru8p 2022-10-06

How to find linearly independent vectors?
There are three vectors:
$a_{1} = (- 1, 1, 0, x) a_{2} = (2, - 3, 1, 2) a_{3} = (1, - 2, 1, - 1)$
How can I find the parameter x so that these vectors are linearly independent? I'm not quite sure how to approach this problem since there are three vectors in $R^{4}$ , so I can't calculate the determinant of the matrix. Any help will be appreciated!

Janessa Benson 2022-10-06

Since $P (X) ⊥ x - P (X)$ , we have $| | x | |^{2} = | | p (x) | |^{2} + | | x - P (x) | |^{2}$ . thus $| | x | |^{2}$ =⟨x,x⟩ = $| | P (x) | |^{2}$ +||x- $P (x) | |^{2}$
Therefore, $| | P (x) | |^{2} \leq | | x | |^{2} . s i n c e | | P (x) | | \geq 0$ and $| | x | | \geq 0, w e h a v e | | P (x) | | \leq | | x | |$
We jump to the conclusion of $| | P (x) | |^{2} \leq | | x | |^{2} .$ because of $| | P (x) | | \geq 0$ and $| | x | | \geq 0.$
but where do we prove that $| | P (x) | | \geq 0$ and $| | x | | \geq 0$ in the first place?

odcinaknr 2022-10-05

The problem is to show that, given $‖ y ‖_{2} = λ^{T} y, ‖ λ ‖_{2} \leq 1$ and $y \neq 0$ , we have $λ = \frac{y}{‖ y ‖_{2}}$
My approach is, $‖ y ‖_{2} = | λ^{T} y | \leq ‖ y ‖_{2} ‖ λ ‖_{2} ⟹ ‖ λ ‖_{2} \geq 1$ which combined with $‖ λ ‖_{2} \leq 1$ gives that $‖ λ ‖_{2} = 1$ . So $λ$ and y are not oppositely aligned, since $‖ y ‖_{2} \neq 0$
Also, $‖ y ‖_{2} = λ^{T} y ⟹ {(\frac{y}{‖ y ‖_{2}} - λ)}^{T} y = 0$ . But since we showed that $λ$ and y are not oppositely aligned, this should mean that the only possibility is $\frac{y}{‖ y ‖_{2}} - λ = 0$ which gives the result.
I feel that there should be a much more straightforward way of seeing the result but can't seem to get there at the moment. Can someone help out?

st3he1d0t 2022-10-05

A plane through the origin is perpendicular to the plane 2x−y−z=5 and parallel to the line joining the points (1,2,3) and (4,−1,2). Find the equation of the plane.

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