Learn Vectors with Examples, Equations, and Practice Problems

Find the possible values of the constant $α$ , given that the vectors $a =< α, 8, 3 α + 1 >$ and $b =< α + 1, α - 1, - 2 >$ are perpendicular to each other
For this question do I have to assume a.b=0?
how can I find the value of constants?
my understanding is: $< α, 8, 3 α + 1 >< α + 1, α - 1, - 2 >= 0$
how do I proceed from here to find the values of constant 𝛼?

ajumbaretu 2022-11-20

I'm trying to complete the following exercise:
Let V be the set of vectors $(x_{1}, x_{2}, x_{3}, x_{4}) \in R^{4}$ such that
$2 x_{1} - 3 x_{2} - x_{3} + x_{4} = 0$
$x_{1} - x_{2} + 2 x_{3} - x_{4} = 0$
Show that V is a subspace of $R^{4}$ and find a basis for V.
I already showed that V is a subspace of $R^{4}$ , but I'm having trouble finding a basis for V. Any help is welcome

Ricky Arias 2022-11-19

Condition on u so that $| \frac{x \cdot u}{u \cdot u} u - x | < | x |$
If $u, x \in R^{n}$ then am I right to assert that the above inequality will hold if u makes an angle $α \in (0, π / 2)$ with span ${x}^{⊥}$ ?
When n=3 it seems quite apparent that $| \frac{x \cdot u}{u \cdot u} u - x | = \cos (α) | x | < | x | .$ I think it's exactly the same in higher dimensions.

Rigoberto Drake 2022-11-19

Prove that: $(\vec{u} \times \vec{v}) \times \vec{w} = (\vec{w} \cdot \vec{u}) \vec{v} - (\vec{v} \cdot \vec{w}) \vec{u}$
I'm trying to solve this question, and my only thoughts of proving it is just to substitute $\vec{u} = (u_{1}, u_{2}, u_{3})$ and same for all vectors and open it up following the dot and cross product rules.
But I was wondering if there's another creative way other than the straight forward way that I'm thinking about.
Appreciate any help

Rhett Guerrero 2022-11-19

Find the possible value/s that the constant c cannot take for the vectors to form a basis in $R^{3}$
I have spent quite a lot of time on the following question:
The vectors
$v_{1} = - c i + 6 j + 2 k$
$v_{2} = 3 i - c j + 2 k$
$v_{3} = - i + 2 j + c k$
form a basis in the 3-dimensional space $R^{3}$ . Find the possible value/s that the constant c cannot take.

Aleah Avery 2022-11-18

Find the angles between the following pairs of vectors
$a = (i + j)$
$b = (2 i - 3 j)$
$a \cdot b = (i + j) \cdot (2 i - 3 j)$
$= 2 (1) - 3 (1) \Rightarrow 2 - 3 = - 1$
$‖ a ‖ = \sqrt{2}$
$‖ b ‖ = i \sqrt{5}$
$\cos θ = \frac{- 1}{\sqrt{2} \sqrt{5}} ⟹ θ = \arccos (\frac{- 1}{\sqrt{2} \sqrt{5}})$
$θ = {108.4}^{\circ}$
My answer is wrong, it is supposed to be ${101.3}^{\circ}$

InjegoIrrenia1mk 2022-11-18

Let $u_{1} = (a_{1}, b_{1}, c_{1}, d_{1})$ and $u_{2} = (a_{2}, b_{2}, c_{2}, d_{2})$ be any vectors in $R^{4}$ Which inner product axioms do not hold with the definition
$⟨ u_{1}, u_{2} ⟩ = a_{1} a_{2} + 2 b_{1} b_{2} - c_{1} c_{2} + 2 d_{1} d_{2}$
Now I believe that symmetry and positivity would hold. It seems that homogeneity could hold, as the scalar doesn't affect the order of multiplication. This would mean additivity is the only one that does not hold. If that is correct, I am having trouble constructing it to show it fails. Cause wouldn't $(a_{1} + a_{2}) w_{1} + 2 (b_{1} + b_{2}) w_{2} - (c_{1} + c_{2}) w_{3} + 2 (d_{1} + d_{2}) w_{4}$ still be an instance of $⟨ u_{1}, w ⟩ + ⟨ u_{2}, w ⟩$ ?

Cortez Clarke 2022-11-18

Find an equation of the plane that passes through the points (1,2,5),(5,4,8), and (2,4,8).
$v_{1} = [1, 2, 5] - [5, 4, 8] = [- 4, - 2, - 3]$
$v_{2} = [2, 4, 8] - [5, 4, 8] = [- 3, 0, 0]$
$v_{1} \times v_{2} = [0, 9, - 6]$
thus,
$a x + b y + c z = d, a = 0, b = 9, c = - 6$
to get d we plug $0 (5) + 9 (4) - 6 (8) = - 12$
Therefore the equation is $9 y - 6 z = - 12$
Right?

Uroskopieulm 2022-11-18

How to find one vector of w in the Euclidean inner product inside a vector space
if $⟨, ⟩$ symbolizes the Euclidean inner product inside the vector space $R^{3}$ , find one vector of w for which
$⟨ w, v_{1} ⟩ + ⟨ w, v_{2} ⟩ = 2 ⟨ w, v_{3} ⟩$
where $v_{1} = [1, 0, 1], v_{2} = [1, 2, 1], v_{3} = [1, 3, 10]$
How can I solve this?

bucstar11n0h 2022-11-17

Angle range when scalar product of vectors less or equal to zero
Given two vectors a and b, what is the possible angle between them when $a b \leq 0$ ?

kituoti126 2022-11-17

How to find parallel tangents for a parametric equation
For the function
$\vec{x} (t) = (\begin{matrix} 2 t + 3 \\ 2 - t \\ t^{3} - 2 t^{2} + t \end{matrix}) t \geq 0$
Are there 2 points $\vec{x} (t_{1})$ , $\vec{x} (t_{2})$ , such that the function’s tangent vectors at these points are parallel to each other? Find such points, or show that none exist.
I know that the derivative of the function is
${\vec{x}}^{'} (t) = (\begin{matrix} 2 \\ - 1 \\ 3 t^{2} - 4 t + 1 \end{matrix})$
and that in order for the tangent vectors to be parallel to each other the functions will equal the same value. However, I am not sure how to go about finding the values.

Jadon Johnson 2022-11-17

Difference between a vector in $R^{1}$ and a scalar
Many people say that they are the same, even I can't find much difference in them except that a vector/ matrix can be multiplied by any scalar, but to multiply it with a vector in $R^{1}$ the vector or matrix should be of the order $1 \times n$ . What's and why is there a difference in this case?

Hayley Mcclain 2022-11-17

Let r be a vector and r be its magnitude. I want to evaluate $\frac{d}{d t} \frac{1}{r}$
My working is
$\begin{aligned} \frac{d}{d t} \frac{1}{r} & = \frac{d}{d r} \frac{1}{r} \frac{d r}{d t} \\ = \frac{- 1}{r^{2}} \frac{d}{d t} (r \cdot \hat{r}) \\ = \frac{- 1}{r^{2}} (\dot{r} \cdot \hat{r} + r \cdot \frac{d \hat{r}}{d t}) \end{aligned}$
I'm pretty sure the answer should be $\frac{- 1}{r^{2}} (\dot{r} \cdot \hat{r})$ but I can't see why $r \cdot \frac{d \hat{r}}{d t} = 0$ . I get that $\hat{r}$ is constant in magnitude, but its direction changes.

Audrey Arnold 2022-11-16

I have the equation $(a + k b)^{2} = c$ where a is a vector, b is a unit vector and k and c are both scalars. By $^{2}$ I mean vector length (is this the correct notation?)
I am trying to solve for k, but I am not sure where to start.
I have noticed when I plot this graph there is sometimes no solution for $k$ (for example if k is less than $a^{2}$ and $b$ is perpendicular to $a$ ), is it possible to solve the lower bound?

Kameron Wang 2022-11-16

Vector equation of a plane passing through $r = (1, 1, - 2)$ , $s = (3, 0, 1)$ , $p = (1, 1, 1)$
For this question I did the cross product so I found rs then rp and using those I did the cross product method to find (a,b,c). After getting a, b, c which was (−3,−6,0). I plugged it into $a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0})$ . This gave me a final equation of $3 x + 6 y = 9$
The answer however is completely different where they said the equation is $(1, 1, - 2) + λ (2, - 1, 3) + μ (0, 0, 3)$
How would I know if my answer was correct from this equation?

Demarion Ortega 2022-11-16

For every vector $w \in R^{3}$ , Aw is in the span of u. Prove this.
I am unsure how to approach this question. Given is
$A = (\begin{matrix} 2 & - 1 & 3 \\ 4 & - 2 & 6 \\ - 2 & 1 & - 3 \end{matrix})$
and
$u = (\begin{matrix} 1 \\ 2 \\ - 1 \end{matrix}) .$
We have to prove for every $w \in R^{3}$ that Aw is in the span of u. So far I have calculated and written Aw as
$A w = (\begin{matrix} 2 w_{1} - w_{2} + 3 w_{3} \\ 4 w_{1} - 2 w_{2} + 6 w_{3} \\ - 2 w_{1} + w_{2} - 3 w_{3} \end{matrix})$
And I know that $s p (u) = {r_{1} + 2 r_{2} - r_{3} ∣ r_{1}, r_{2}, r_{3} \in R}$
But I do not really know where to go from here. Any help or guidance is appreciated.

Alberto Calhoun 2022-11-15

Prove that the volume of the tetrahedron $A B C D$ is $\frac{1}{6} A B \cdot C D \cdot E F \sin x$ where $E F$ is the shortest distance between $A B$ and $C D,$ and x is the angle between these two lines. I have attempted the proof using vector methods, but have not got very far. Any help would be appreciated.

odcizit49o 2022-11-15

Find the area of a triangle formed by vectors $𝑥 ⃗$ and $𝑦 ⃗$ , if $𝑥 ⃗$ = $\vec{A} + 2 \vec{B}$ , $𝑦 ⃗$ = $2 \vec{A} - \vec{B}$ where | $\vec{A}$ | = 3, | $\vec{B}$ | = 4, and the angle between $\vec{A}$ and $\vec{B}$ is $π / 6$
I can't figure out how to find it without vectors components.

Rhett Guerrero 2022-11-15

Given B =
$(\begin{matrix} 1 & - 1 & 1 \\ 1 & 1 & 3 \end{matrix})$
determine the general solution of the homogeneous system Bx=0. This describes the intersection of two objects (e.g. a line or a plane). Determine and find the Cartesian equation of this object.
I have solved the homogeneous equation being x $= t (- 2, - 1, 1)$ but how do I determine what two objects this intersection represents along with their Cartesian equation?

anraszbx 2022-11-15

First of all $v, w, z$ are in $R^{3}$ as vectors, and there are functions $G (v, w, z) = (v \times w) \times z$ , this is cross product. So I know this function is linear in the 3 arguments, so for example
$G (x + y) = G (x) + G (y)$ and a is a constant. First of all what is means to be linear in second argument? And, also, prove this G function is linear in the second argument.

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