Learn Vectors with Examples, Equations, and Practice Problems

The name directional suggests they are vector functions. However, since a directional derivative is the dot product of the gradient and a vector it has to be a scalar. But, in my textbook, I see the special case of the directional derivatives $F_{x} (x, y, z)$ and $F_{y} (x, y, z)$ being treated as vectors. I want a clarification for this.

Ty Moore 2022-11-02

Let $z \in R^{n}, r > 0$ , and $ϵ \in (0, 2]$ . Prove that if $x, y \in \bar{B} (z, r)$ such that $‖ x - y ‖ \geq ϵ r,$ then $‖ z - \frac{x + y}{2} ‖ \leq r \sqrt{1 - \frac{ϵ^{2}}{4}}$ . So far I have:
$‖ z - \frac{x + y}{2} ‖ = \sqrt{⟨ z - \frac{x + y}{2}, z - \frac{x + y}{2} ⟩}$
Then
$⟨ z - \frac{x + y}{2}, z - \frac{x + y}{2} ⟩ = ⟨ z - x, z - \frac{x + y}{2} ⟩ + ⟨ \frac{x - y}{2}, z - \frac{x + y}{2} ⟩$
$= ⟨ z - x, z - y ⟩ + ⟨ z - x, \frac{y - x}{2} ⟩ + ⟨ \frac{x - y}{2}, \frac{y - x}{2} ⟩ + ⟨ \frac{x - y}{2}, z - y ⟩$
$= ⟨ z - x, z - y ⟩ + ⟨ \frac{x - y}{2}, x - y ⟩ - 1 / 4 ‖ x - y ‖^{2}$
$= ⟨ z - x, z - y ⟩ + 1 / 2 ‖ x - y ‖^{2} - 1 / 4 ‖ x - y ‖^{2}$
$\leq ‖ z - x ‖ ‖ z - y ‖ - \frac{(ϵ r)^{2}}{4} + 1 / 2 ‖ x - y ‖^{2}$
$\leq r^{2} - \frac{(ϵ r)^{2}}{4} + 1 / 2 ‖ x - y ‖^{2} = r^{2} (1 - \frac{ϵ^{2}}{4}) + 1 / 2 ‖ x - y ‖^{2}$
I think I'm on the right track but I keep getting the $1 / 2 ‖ x - y ‖^{2}$ and I don't know what to do.

oopsteekwe 2022-11-01

Let x, y, z, t are real numbers. x+y+z+t=6 and $\sqrt{1 - x^{2}} + \sqrt{4 - y^{2}} + \sqrt{9 - z^{2}} + \sqrt{16 - t^{2}} = 8$ Find each value of them as a number.

varsa1m 2022-11-01

Let $u, v \in R^{n}$ . Prove that if
$‖ u + t v ‖ \geq ‖ u ‖$
for all $t \in R$ , then $u \cdot v = 0$ (vectors u and v are perpendicular).

oopsteekwe 2022-10-31

$(\vec{a} - \vec{b})$ perpendicular $(6 \vec{a} + \vec{b})$
$6 {\vec{a}}^{2} - 5 \vec{a} \vec{b} - {\vec{b}}^{2} = 0$
$12 - 5 \vec{a} \vec{b} - 3 = 0$
$5 (6 \cos θ) = 9$
But the answer is $\frac{π}{3}$ . Where am i wrong?

duandaTed05 2022-10-30

If a subset S in a vector space V is linearly dependent & $T \in$ L(V, W), then T(S) is linearly dependent in $W$
Is $L (V, W)$ the linear combination of two spaces V and W? If so, what is T(S)? I am having some trouble on getting started on this problem.

Taniya Melton 2022-10-30

I have a function $s^{T} x_{i} x_{j}^{T} s$ where $s \in R^{d}$ and $x \in R^{d}$ which is computed at every index of matrix which means $K_{i j} = s^{T} x_{i} x_{j}^{T} s$ with $K_{i j} \in R^{n x n}$ . Now if i take any vector $v \in R^{n}$ then how to multiply that vector with matrix with matrix in this form $v^{T} . K . v$ . What I have done so far is
$\sum_{i, j} z_{i} . (s_{i} . x_{i} . x_{j} . s_{j}) . z_{j}$
Is it possible to complete the square of the above term with the one given below?
$\sum_{i} | | z_{i} . s_{i} . x_{i} | |^{2}$

erwachsenc6 2022-10-29

Prove for nonzero scalars a and b such that
$s p a n {v_{1}, v_{2}} = s p a n {a v_{1}, b v_{2}}$
My try:
$s p a n {v_{1}, v_{2}} = c_{1} v_{1} + c_{2} v_{2} = c_{1}^{'} (a v_{1}) + c_{2}^{'} (b v_{2}) = s p a n {a v_{1}, b v_{2}}$
by splitting contants. Is it correct to show ? then what is the role of non zero scalar word here ?

Alexander Lewis 2022-10-29

How do you prove this if x is a block vector with two vector elements, $[\begin{matrix} a \\ b \end{matrix}]$ where a and b are vectors of size n and m respectively?
$\begin{aligned} ‖ x ‖ = \sqrt{‖ a ‖^{2} + ‖ b ‖^{2}} = ‖ [\begin{matrix} ‖ a ‖ \\ ‖ b ‖ \end{matrix}] ‖ \end{aligned}$

Ralzereep9h 2022-10-28

Let C be the part of the plane curve defined by
$y^{2} = x^{3} - x$
between
$(\frac{- 1}{\sqrt{3}}, \sqrt[4]{\frac{4}{27}})$
and
$(0, 0)$
oriented from left to right. How would I calculate
$\int_{C} y^{2} \vec{i} + (2 x y + 4 y^{3} e^{y^{4}}) \vec{j} d s$
I have already found that the vector field is conservative, I'm just not sure how to proceed from there.

Aydin Jarvis 2022-10-28

Find the points where the line given by x=7t+9, y=8t−7, and z=−7t−2 intersect the xy, xz, and yz planes.
Attempted solution:
This seems as though a very simple problem to me. Take, for example, finding the intersection with the xz plane. It must be that y=0. I substitute y=0 to obtain $t = \frac{7}{8}$ , then $x = \frac{121}{8}$ and $z = - \frac{65}{8}$ . The point of intersection is $(\frac{121}{8}, 0, - \frac{65}{8})$
It seems, however, that my solution is wrong, which completely baffles me.
What is the correct approach?

Maribel Mcintyre 2022-10-28

In 3d space, is there any way to project a vector onto a plane, but along the UP direction (0,1,0) instead of the plane normal? If so, how do I do that and what is it called?

Antwan Perez 2022-10-27

If I have 3 vectors, a, b and c in 3D, I can check if they fulfill $c = α a + β b$ (i.e. if they lie in a 2D plane) for some real parameter $α$ and $β$ by checking if $(a \times b) \cdot c = 0$ . If I have 3 vectors in n dimensions, is there a similar, general formula to check if $c = α a + β b$ ?

Ralzereep9h 2022-10-26

Consider the triangle ABC and the midpoint A′ of the side [BC].Show that $4 {\vec{A A^{'}}}^{2} - {\vec{B C}}^{2} = 4 \vec{A B} \cdot \vec{A C}$
I have computed that $\vec{A A^{'}} = \frac{\vec{A B} + \vec{A C}}{2}$ but I don't know what to do next. Some tips please?

Hugo Stokes 2022-10-25

Determine the length of the projection of PQ onto the x-axis, with position vectors of P <3,4> and Q <7,8>
My working out so far:
PQ=PO+OQ
=−(3i+4j)+(7i+8j)
=4i+4j
Since PQ is being projected onto the x-axis, do you choose a random vector lying on the x-axis, like a unit vector (1i+0j) to calculate the scalar projection with the formula?
Or the answer key provided suggests that the scalar is just the x-component of PQ - so in this case it would just be 4 since that is the i component, but I don't understand why. Could someone please explain this to me?

beefypy 2022-10-24

Problem:For vectors u and v, we have $p = {proj}_{v} (u)$ . If $‖ u ‖ = 11$ and $‖ p ‖ = 6$ , find $p \cdot u$
I don't understand how to do this problem because, to me, it seems like there isn't enough information, but I know that there is. I know that any vector and its projection either form the hypotenuse and a leg of a right triangle, or that they are equal. So I can figure out the cosine of the angle between u and p. But, I don't know how exactly I would do that?

Mattie Monroe 2022-10-23

I am trying to evaluate $E [sign ⟨ v, z ⟩]$ for $v \in R^{n}$ fixed and $z_{i} \sim N (0, 1) \forall i \in [n]$
The sign part is what is confusing me. Clearly,
$E [sign ⟨ v, z ⟩] = E [sign \sum_{i = 1}^{n} v_{i} z_{i}] = P [\sum_{i = 1}^{n} v_{i} z_{i} > 0] - P [\sum_{i = 1}^{n} v_{i} z_{i} < 0] .$
But I don't know how to simplify further (i.e. to get the expectation just on the $z_{i}$ ). I am wondering how to pass the sign operator through the expectation.

Kymani Hatfield 2022-10-23

We are given two vectors $(a, b), (c, d) \in R^{2}$ which lie in the same one dimensional subspace, and at least one of a,b,c,d is nonzero. How can we find a single vector in terms of a,b,c,d which spans the subspace containing the two original vectors?

cousinhaui 2022-10-23

I have the vector $\vec{c}$ that is:
$\vec{c} = \frac{\sum_{i = 1}^{n} m_{i} {\vec{r}}_{i}}{\sum_{i = 1}^{n} m_{i}}$
where ${\vec{r}}_{i}$ is a vector and $m_{i}$ is a scalar
I need to proof the folowwing equality for any vector $\vec{r}$
$\sum_{i = 1}^{n} m_{i} | \vec{r} - {\vec{r}}_{i} |^{2} = \sum_{i = 1}^{n} m_{1} | {\vec{r}}_{i} - \vec{c} |^{2} + m | \vec{r} - \vec{c} |^{2}$
and i known that $m = \sum_{i = 1}^{n} m_{i}$
I try to replace the vector $\vec{c}$ in the equality but but i get confused with the vector algebra.

c0nman56 2022-10-23

Determine coefficient $λ$ so vectors p and q are mutually perpendicular
$p = λ a + 17 b$
q=3a-b
|a|=2 , |b|=5 , angle between a & b vector is $120^{\circ}$
I got for vectors a and b
$\vec{a} = (- 1, \sqrt{3}) \vec{b} = (\frac{- 5}{2}, \frac{5 \sqrt{3}}{2})$
how do I find coefficient $λ$ ?

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