Learn Vectors with Examples, Equations, and Practice Problems

Given two non-zero vectors $\vec{a}$ and $\vec{b}$ such that | $\vec{a}$ + $\vec{b}$ | = | $\vec{a}$ - $\vec{b}$ |, find the value of $\vec{a}$ • $\vec{b}$

jorgejasso85xvx 2022-11-08

Is it true to write that $\hat{M} (\vec{a} \times \vec{b}) = \hat{M} \vec{a} \times \hat{M} \vec{b}$ ? If so, how can this be proven?

Jefferson Booth 2022-11-08

$C^{n}$ is a C-vector space with a natural structure of $R$ -vector space. Then, what is the dimension of $C^{n}$ as $R$ -vector space? I'm struggling so hard with this question, I do not even know how to start. Can somebody help me please.

Anton Huynh 2022-11-07

Show linear transformation is not one-to-one
$T (x, y, z) = [\begin{array}{ccc} x - 2 z \\ - 3 x + y + 3 z \\ x - y + z \end{array}]$
I've taken the matrix of coefficients:
$A = [\begin{matrix} 1 & 0 & - 2 \\ - 3 & 1 & 3 \\ 1 & - 1 & 1 \end{matrix}]$
And reduced it to row-echelon form as follows:
$A = [\begin{matrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \end{matrix}]$
But, I'm not sure where to go from here in proving that this transformation is not one-to-one.
I understand that a one-to-one transformation has a pivot in every column, but I'm not sure what this means and how it applies to this matrix.
Where can I go from here in proving this?

figoveck38 2022-11-07

Suppose I have an $n \times n$ matrix A with positive real entries. Is it possible to find an $n \times 1$ vector v such that
$v v^{T} = A$
where $v^{T}$ is the transpose of $v$ ?
Is it possible to use singular value decomposition or any technique to achieve this?

Karley Castillo 2022-11-06

Let x and y be two real vectors of length n. Let subscripts i and j denote the i-th and j-th elements of a vector.
How can we prove
$| x_{i} - x_{j} | \geq | y_{i} - y_{j} | - 2 | | x - y | |_{\infty}$
for any i and j. Here, $| | x | |_{\infty} = max_{1 \leq i \leq n} | x_{i} |$

Brenda Jordan 2022-11-06

Recently I came across a topic " total differential" which comes with a result
$d f = \frac{\partial f}{\partial x} d x + \frac{\partial f}{\partial y} d y$
As much I learned in multivariable calculus this can be simplified as $\nabla f \cdot [\begin{array}{l} d x \\ d y \end{array}]$ which graphically means taking directional Derivative in diraction of $[\begin{array}{l} d x \\ d y \end{array}]$ But is it Makes any sense ?

limunom623 2022-11-05

Let an angle between the vectors a and b be $\frac{2 π}{3}$ . If $| b | = 2 | a |$ , and the vectors $a + x b$ and a−b are at right angles, then the value of x is equal to $\frac{2}{3} / \frac{2}{5} / \frac{1}{3} / \frac{1}{5}$ ?
My attempt: $b^{2} = 4 a^{2}$ . Also,
$a^{2} + \frac{| a | | b |}{2} - \frac{x | a | | b |}{2} - b^{2} = 0$
$⟹ a^{2} + \frac{| a | * 2 | a |}{2} - \frac{x | a | * 2 | a |}{2} - 4 a^{2} = 0$
$⟹ 1 + 1 - x - 4 = 0$
$⟹ x = - 2$
But it doesn't match with the options.

linnibell17591 2022-11-05

The vectors $(\binom{3}{2})$ and $(\binom{- 4}{1})$ can be written as linear combinations of u and w:
$\begin{aligned} (\binom{3}{2}) & = 5 u + 8 w \\ (\binom{- 4}{1}) & = - 3 u + w . \end{aligned}$
The vector $(\binom{5}{- 2})$ can be written as the linear combination au+bw. Find the ordered pair (a,b).
I've tried to eliminate u by multiplying the first equation by 3, the second equation by 5, then adding, but it only leads to $w = (\binom{- \frac{11}{29}}{\frac{11}{29}})$ . I feel like the algebra from here would be too complicated for what the people who wrote the problem were intending, so perhaps I'm going down the wrong path. Would there instead be a convenient way to manipulate the terms to eventually get $(\binom{5}{- 2})$ on the LHS?

Demarion Ortega 2022-11-05

Question: Find the vector sum:
$\vec{A B} + \vec{C E} - \vec{C B} - \vec{K E} + \vec{M P} - \vec{K P}$
I have approached the problem like this:
$\vec{A B} + \vec{B C} + \vec{C E} - \vec{K E} + \vec{M P} - \vec{K P} = \vec{A C} + \vec{C E} + \vec{E K} + \vec{M P} - \vec{K P} = \vec{A E} + \vec{E K} + \vec{M P} - \vec{K P} = \vec{A K} + \vec{M P} - \vec{K P} = \vec{A K} + \vec{M P} + \vec{P K} = \vec{A K} + \vec{M K}$
How do I simplify $\vec{A K} + \vec{M K}$ ?

Kareem Mejia 2022-11-05

A plane $π$ contains the line $r = i + 3 j - k + λ (2 i - j + k)$ . If $π$ is parallel to 3i - 4j, how to find the vector equation of $π$ ?
I know that parallel planes have the same perpendicular vector. But I don’t seem to see how to find any normal vectors to the plane. I can’t cross product two lines. How should I solve this question?

Jenny Roberson 2022-11-05

If x is a $n \times 1$ vector, are the entries of diag( $x x^{T}$ ) given by $x^{T} x$ ?

Messiah Sutton 2022-11-04

I am trying to measure/estimate the angular acceleration of an object $\dot{ω}$ from a measurement of it's acceleration (using an accelerometer) $^{i} a_{m}$ . As far as I understand, the accelerometer will measure a linear acceleration due to rotation of the object.
From rigid body kinematics, the following relation is know
$\begin{aligned} ^{i} a_{m} & =^{i} a_{l} +^{i} {\dot{ω}}_{i} \times^{i} X_{S_{m}} +^{i} ω_{i} \times (^{i} ω_{i} \times^{i} X_{S_{m}}) \end{aligned}$
Assuming I know everything in this equation except for $\dot{ω}$ , I would like to estimate $\dot{ω}$ .
Unfortunately, the equation cannot directly be solved to $\dot{ω}$ since $\dot{ω}$ is in a cross product with a vector.
Are there any mathematical tools that can help me estimate $\dot{ω}$ given the relation I described above?

Jonas Huff 2022-11-04

Consider two vectors $\vec{a}$ , $\vec{b}$ whose components are not known. $| \vec{a} | = 3$ , $| \vec{b} | = 2$ and an angle between them is $120^{\circ}$ . Without assuming any particular components, calculate the magnitude of the vector product $| (\vec{a} - 3 \vec{b}) \times (2 \vec{a} + \vec{b}) |$
I understand that when I just want to find the vector product given the above information I would use:
$| \vec{a} \times \vec{b} | = | \vec{a} | | \vec{b} | \sin (θ) .$
When I do this I end up with $| \vec{a} \times \vec{b} | = 3 \sqrt{3}$ . I am not sure whether I am supposed to do this and if I am where I go on from here to calculate $| (\vec{a} - 3 \vec{b}) \times (2 \vec{a} + \vec{b}) |$

klasyvea 2022-11-04

Show that (1,0,1),(−2,1,4),(0,3,1) forms the basis for $\in R^{3}$
I want to show that $[\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} - 2 \\ 1 \\ 4 \end{matrix}], [\begin{matrix} 0 \\ 3 \\ 1 \end{matrix}]$ forms the basis for $\in R^{3}$ . First I tried to show that the vectors are linearly independent
$α_{1} [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] + α_{2} [\begin{matrix} - 2 \\ 1 \\ 4 \end{matrix}] + α_{3} [\begin{matrix} 0 \\ 3 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
But I tried to continue from this on wards but I get that $α_{3} = 18 α_{3}$ , which doesn't make sense so I'm a bit lost.
$α_{1} [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] + α_{2} [\begin{matrix} - 2 \\ 1 \\ 4 \end{matrix}] + α_{3} [\begin{matrix} 0 \\ 3 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

vedentst9i 2022-11-04

Let $x, y \in R^{n}$ . Let A be a $n \times n$ positive-definite symmetric matrix. Is there a general formula for $x^{T} A x \cdot y^{T} A y$ ?
For example, let $x = [\begin{matrix} 2 \\ 2 \end{matrix}], y = [\begin{matrix} 3 \\ 3 \end{matrix}], A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
Then $x^{T} A x \cdot y^{T} A y = 8 \cdot 18 = 144$
This is equal to $2 (x \circ y)^{T} A (x \circ y)$ where $(x \circ y) = [\begin{matrix} 6 \\ 6 \end{matrix}]$ is the Hadamard product of x,y.
It seems that $x^{T} A x \cdot y^{T} A y = 2 (x \circ y)^{T} A (x \circ y)$ also works for $x = [\begin{matrix} 5 \\ 5 \end{matrix}], y = [\begin{matrix} 7 \\ 7 \end{matrix}]$ . Is this true in the general case, and if so, how do I prove it? If not, how can I find and prove a general formula?

vedentst9i 2022-11-04

Given two vectors - $u = (\frac{1}{n}, \frac{1}{n}, . . ., \frac{1}{n})$ of the uniform distribution on { $1,, . . ., n$ }, and a stochastic vector $p \in R^{n}$ , I need to show that:
$‖ p - u ‖_{2} \leq 1$
I started with the following step:
$‖ p - u ‖_{2} = \sqrt{(p_{1} - \frac{1}{n})^{2} + (p_{2} - \frac{1}{n})^{2} + (p_{3} - \frac{1}{n})^{2} + . . . + (p_{n} - \frac{1}{n})^{2}} \leq \sqrt{(1 - \frac{1}{n})^{2} + (\frac{1}{n})^{2} + (\frac{1}{n})^{2} + . . . + \frac{1}{n})^{2}}$
Is this step correct? If not, what am I missing?

Davirnoilc 2022-11-04

I just started reading about vectors. For the vector $[\begin{matrix} x_{1} \\ x_{2} \\ 0 \end{matrix}]$ (where $x_{1}, x_{2} \neq 0$ ) I know we have $[\begin{matrix} x_{1} \\ x_{2} \\ 0 \end{matrix}] \in R^{3}$ . But because the vector is on the xy-plane, I wonder is it correct to say $[\begin{matrix} x_{1} \\ x_{2} \\ 0 \end{matrix}] \in R^{2}$ ? and why?

clealtAfforcewug 2022-11-03

Given $A = (a_{1}, a_{2}), B = (b_{1}, b_{2}), C = (c_{1}, c_{2})$ , is there a simple formula to express the radius of the circumcircle of ABC?
Note that you could compute the radius from the sidelengths as $\frac{a b c}{\sqrt{(a + b + c) (- a + b + c) (a - b + c) (a + b - c)}}$ , but I'm really hoping that there's something simpler than that. If it helps, assume C=(0,0).

klasyvea 2022-11-02

What is the fastest way of calculating whether vectors are linearly independent?
Is the following family $(u_{1}, u_{2}, u_{3})$ linearly independent?
$u_{1} := (\begin{matrix} 1 \\ 3 \\ 5 \\ - 1 \end{matrix}), u_{2} := (\begin{matrix} 1 \\ - 1 \\ - 3 \\ 3 \end{matrix}), u_{3} := (\begin{matrix} 3 \\ 2 \\ 1 \\ 4 \end{matrix})$
So, do I have to form a matrix U with columns $u_{1}$ , $u_{2}$ and $u_{3}$ and use gaussian elimination to see whether in every column is a pivot or is there a faster way?

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