Learn Vectors with Examples, Equations, and Practice Problems

If $(b + c) x + (c + a) y + (a + b) z = k = (b - c) x + (c - a) y + (a - b) z$ then what will be the equation of the straight line passing through origin and parallel to given line ?
I tried to relate the direction ratios of the given line and the new line as they are parallel but the problem is that line is passing through origin so I am ending up getting all zeros. Please help. Any hint will do .

imchasou 2022-09-23

What is the magnitude (length) of vector µv for the following $μ$ ? a) $μ$ = 0 b) $μ$ = length of v c) $μ$ = 1/length of v $μ$ = 1/length of v squared
What is $μ v$ ? what is $μ$ ?

Freddy Chaney 2022-09-23

Let $Z \in R^{n}$ , $α \in R^{n}$ , $γ > 0$ . Now I would like to find a solution for $α$ of the following equation,
$\begin{aligned} Z - α = γ \frac{α}{‖ α ‖_{2}} . \end{aligned}$
I am totally unfamiliar with such an equation. So I already get stuck here. Any help is really appreciated (or sources/references).

zaviknuogg 2022-09-22

Verify Gauss divergence theorem of $F = x i + y j + z k$ over the sphere $x^{2} + y^{2} + z^{2} = a^{2}$

Janet Hart 2022-09-22

Given a vector $v \in R^{n}$ then show that we can express $\sum_{k} ω_{k} v_{k}^{2}$ as a matrix product of the form $v^{T} M v$ . Give an expression for M in terms of $ω = [ω_{1} . . . ω_{n}]^{T}$
I understand that here, the product of a row vector, matrix, and column vector (in that order) is a scalar. However, how do I write M in terms of $ω = [ω_{1} . . . ω_{n}]^{T}$ ?

Kelton Molina 2022-09-22

We know that the the norm of a vector is $‖ \vec{u} ‖ = \sqrt{u_{1}^{2} + u_{2}^{2} + . . . + u_{n}^{2}}$ , where $\vec{u} \in R^{n}$ . How does the d x 1 vector look like using the definition of norm, and what does $R^{n}$ really means? Or what is $R^{n}$ for the d x 1 vector?

furajat4h 2022-09-22

For some context, I am learning about the cross product, and the matrix we used as an example was: $V = [\begin{matrix} \vec{v_{1}} \\ \vec{v_{2}} \\ \vec{v_{3}} \end{matrix}]$ which I am told is a $3 \times 3$ matrix. But since these vectors are in 3 dimensions, wouldn't they also be column vectors? How do we get to a $3 \times 3$ square from that?

hotonglamoz 2022-09-22

I understand how to solve this problem: i.e. take the unit vector of a, then multiply by our given magnitude (in this case $\sqrt{2}$ ) which gives: $\frac{\sqrt{2}}{5} (3 i + 4 j)$
I'm confused as to what the mathematical logic behind this, how would you explain why we're doing this process?

waldo7852p 2022-09-21

Show that $\vec{a} \cdot \vec{b} = \frac{1}{4} | \vec{a} + \vec{b} |^{2} - \frac{1}{4} | \vec{a} - \vec{b} |^{2}$
I have tried:
$\vec{a} \cdot \vec{b} = \frac{1}{4} | \vec{a} + \vec{b} |^{2} - \frac{1}{4} | \vec{a} - \vec{b} |^{2} \vec{a} \cdot \vec{b} = \frac{1}{4} (| \vec{a} |^{2} + 2 | \vec{a} | | \vec{b} | + | \vec{b} |^{2}) - \frac{1}{4} (| \vec{a} |^{2} - 2 | \vec{a} | | \vec{b} | + | \vec{b} |^{2}) \vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} |$
However it seems that the equation only holds true when $\vec{a}$ and $\vec{b}$ are collinear.
Is that true?

Corbin Bradford 2022-09-20

Let X be a $n \times 1$ column vector, whose components are all 0 except for the ith component, and A is a $m \times n$ matrix, what is AX ?
$X = (\begin{matrix} 0 \\ ⋮ \\ 1 \\ ⋮ \\ 0 \end{matrix}) and A = (\begin{matrix} a_{11} & \dots & a_{1 n} \\ ⋮ & ⋮ \\ a_{m 1} & \dots & a_{m n} \end{matrix})$
I've tried several examples for X and A, and I've noticed that there is a relation between the position of the ith component of X (which is 1), and AX, but I can't like really make the generalization.

sombereki51 2022-09-19

If two vectors $A = a i - j - 4 k$ and $B = 2 i - b j + 3 k$ and 3A=2B, what are the values of a and b?
I have tried to solve it by first calculating 3A and 2B. As these two vectors are equal their magnitudes should be equal too which gives me the equation
$9 a^{2} + 153 = 4 b^{2} + 52 ⟹ 9 a^{2} - 4 b^{2} = - 101 . . . . . . . (1)$
the answer in the book is given as $a = 4 / 3$ and $b = 3 / 2$ . I think these two values should satisfy equation (1). But it doesn't. Can anyone verify if I am doing the math in the wrong way or the answers are wrong.

Melina Barber 2022-09-19

Trying to prove:
Show that there are no vectors u and v such that $‖ u ‖ = 1$ , $‖ v ‖ = 2$ , and $u \cdot v = 3$
Don't know where to go from here:
$‖ u ‖ = 2 ⟹ ‖ u ‖^{2} = 4 ⟹ u \cdot u = 1$
$‖ v ‖ = 1 ⟹ ‖ v ‖^{2} = 1 ⟹ v \cdot v = 4$
Not sure if this is the right direction to take, but we have:
$\begin{aligned} u \cdot v = v \cdot v - u \cdot u = 3 \\ = (v + u) \cdot (v - u) \end{aligned}$

ghulamu51 2022-09-19

I have a set of nonzero vectors in $R^{n}$ , where for each vector, its nonzero elements have the same magnitude. For example, when $n = 4$ , $(1, 0, 1, 0)$ and $(- 3, 3, 0, 3)$ are in this set, while $(1, 2, 3, 2)$ and $(0, 0, 0, 0)$ are not.
The formal notation I came up for it is
${x \in R^{n} ∣ x \neq 0 and x_{i} \in {0, k, - k} for some k \in R},$
but I'm not sure if the "for some k" part is correct. Should it be "for all k" instead?

Madelynn Winters 2022-09-18

Given vectors $\vec{A B}, \vec{D C},$ and $\vec{B C}$ , simplify $\vec{A B} - \vec{D C} + \vec{B C}$
Here's what I have done:
$\vec{A B} - \vec{D C} + \vec{B C}$
$\vec{A B} + \vec{C D} + \vec{B C}$
By Triangle law, $\vec{A B} + \vec{C D} = \vec{A D}$
Plugging back in we get:
$\vec{A D} + \vec{B C}$
Again using Triangle law: $\vec{A D} + \vec{B C} = \vec{A C}$
But, my online grader said it was wrong. Please help!

Dymnembalmese2n 2022-09-17

I have three points with coordinates: $A (5, - 1, 0), B (2, 4, 10)$ and $C (6, - 1, 4)$
I have the following vectors $\vec{C A} = (- 1, 0, - 4)$ and $\vec{C B} = (- 4, 5, 6)$
To find the area of the triangle I used the dot product between these vectors to get the angle and then applied the formula $A = 0.5 a b \sin C$ to find the area of the triangle which gave me 15.07(2dp).
However in the given solutions the answer is given as $(3 * \sqrt{(} 102)) / 2$
I think they have used the trig identity $\cos^{2} (θ) + \sin^{2} (θ) = 1$ to find the value of $\sin (θ)$ rather than $\arccos (θ)$ to find the angle ACB. However I don't understand why there would be such a discrepancy between the two answers; one using $\arccos$ and the other using the trig identity.

2k1ablakrh0 2022-09-17

I keep hearing that if you have an $R^{3}$ space you need 3 elements to fill the space.
So, let's consider a column vector. I understood 3 elements to mean = 3 column vectors with 3 elements each. As you can't span $R^{3}$ with JUST one or two vectors even if they each have 3 elements. Correct?
Could you technically somehow take 2 column vectors with 4 elements to span $R^{3}$ ? I'm guessing not.

Quinlan7g 2022-09-17

Question: Consider the vectors u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1) and u4 = (0, 0, 0, 1). Write down an arbitrary vector (a1, a2, a3, a4) ∈ $R^{4}$ as a linear combination of u1, u2, u3 and u4.
Can I just do $(\begin{matrix} a 1 \\ a 2 \\ a 3 \\ a 4 \end{matrix}) = k \cdot u 1 + b \cdot u 2 + c \cdot u 3 + d \cdot u 4$ ? Is that it? What is the question trying yo highlight with all those ones?

traffig75 2022-09-17

Is it true that $(\vec{w} - \frac{1}{2} \vec{u}) ⊥ (2 \vec{w} - \vec{u}) \forall \vec{u}, \vec{w}$ ?
I think is false. Let $\vec{u} = (1, 0, 0)$ and $\vec{w} = (0, 1, 0)$ two vectors. Then
$(\vec{w} - \frac{1}{2} \vec{u}) \cdot (2 \vec{w} - \vec{u}) = (- \frac{1}{2}, 1, 0) \cdot (- 1, 2, 0) = \frac{1}{2} + 2 = \frac{5}{2} \neq 0.$
Is it correct?

Haiphongum 2022-09-16

Let $U, W \subseteq V$ , vector subspaces then:
1) $U \subseteq W ⟹ W^{⊥} \subseteq U^{⊥}$
2) $(U + W)^{⊥} = U^{⊥} \cap W^{⊥}$

joguejaseg 2022-09-16

I have the following triangle coordinates:
$A = (0, 1 / 2, 3 / 2)$
$B = (1, 1, 0)$
$C = (1, 0, 1)$
How can I calculate the perimeter of the triangle ABC ?
Should I add the vertices of the triangle like this?
$A B : \sqrt{(0 - 1)^{2} + (1 / 2 - 1)^{2} + (3 / 2 - 0)^{2}}$
$B C : \sqrt{(1 - 1)^{2} + (1 - 0)^{2} + (0 - 1)^{2}}$
$C A : \sqrt{(1 - 0)^{2} + (0 - 1 / 2)^{2} + (1 - 3 / 2)^{2}}$
And how can I calculate the value of the internal angle of C?

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