Find the vector and parametric equations for the line segment connecting P to Q.

P(0, - 1, 1), Q(1/2, 1/3, 1/4)

sodni3
2021-05-29
Answered

Find the vector and parametric equations for the line segment connecting P to Q.

P(0, - 1, 1), Q(1/2, 1/3, 1/4)

You can still ask an expert for help

Szeteib

Answered 2021-05-30
Author has **102** answers

Vector equation of a line segment joining the points with position vectors ${r}_{0}$ and ${r}_{1}$ is

$r=(1-t){r}_{0}+t{r}_{1}$

Where$t\in [0,1]$

Substitute${r}_{0}=<0,-1,1>$ and ${r}_{1}=<\frac{1}{2}.\frac{1}{3},\frac{1}{4}>$

$r(t)=(1-t)(0,-1,1)+t<\frac{1}{2},\frac{1}{3},\frac{1}{4}>$

$r(t)=<0,-1+t,1-t>+<\frac{t}{2},\frac{t}{3},\frac{t}{4}>$

$r(t)=<\frac{t}{2},-1+\frac{4t}{3},1-\frac{3t}{4}>$

Where$t\in [0,1]$
The parametric equations for the line segment are

$x=\frac{t}{2},y=-1+\frac{4t}{3},z=1-\frac{3t}{4}$

Where$t\in [0,1]$

Where

Substitute

Where

Where

asked 2021-02-11

Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in $M}_{2\times 4$ with the property that FA = 0 (the zero matrix in ${M}_{3\times 4})$ . Determine if H is a subspace of $M}_{2\times 4$

asked 2021-05-17

Find the scalar and vector projections of b onto a.

$a=(4,7,-4),b=(3,-1,1)$

asked 2021-05-29

Which of the following expressions are meaningful? Which are meaningless? Explain.

a)$(a\cdot b)\cdot c$

$(a\cdot b)\cdot c$ has ? because it is the dot product of ?.

b)$(a\cdot b)c$

$(a\cdot b)c$ has ? because it is a scalar multiple of ?.

c)$|a|(b\cdot c)$

$|a|(b\cdot c)$ has ? because it is the product of ?.

d)$a\cdot (b+c)$

$a\cdot (b+c)$ has ? because it is the dot product of ?.

e)$a\cdot b+c$

$a\cdot b+c$ has ? because it is the sum of ?.

f)$|a|\cdot (b+c)$

$|a|\cdot (b+c)$ has ? because it is the dot product of ?.

a)

b)

c)

d)

e)

f)

asked 2022-08-14

The line is perpendicular to the plane 2x–3y+z–w=7 and passes the origin. Find parametric form vector form

This is my solution:

The line perpendicular to the plane implies being parallel to the normal vector. n=[2,−3,1,−1]

vector form: [x,y,z,w,v]=[0,0,0,0]+t[2,−3,1,−1]

parametric form: x=2t,y=−3t,z=t,w=−t,v=0

right?

This is my solution:

The line perpendicular to the plane implies being parallel to the normal vector. n=[2,−3,1,−1]

vector form: [x,y,z,w,v]=[0,0,0,0]+t[2,−3,1,−1]

parametric form: x=2t,y=−3t,z=t,w=−t,v=0

right?

asked 2022-09-25

Let $\lambda $ be an eigenvalue of A, such that no eigenvector of A associated with $\lambda $ has a zero entry. Then prove that every list of n−1 columns of $A-\lambda I$ is linearly independent.

asked 2022-07-14

The following set of points is a lattice in ${\mathbb{R}}^{n}$:

${D}_{n}=$ $({v}_{1},...,{v}_{n})|{v}_{1},...,{v}_{n}\in \mathbb{Z}$ and ${v}_{1}+...+{v}_{n}$ is even}

Find vectors ${x}_{1},...,{x}_{k}$ such that ${D}_{n}=Int({x}_{1},...,{x}_{k})$

My solution:

Firstly, I know that if ${x}_{1},...,{x}_{k}\in {\mathbb{R}}^{n}$, then we define $Int({x}_{1},...,{x}_{k})={t}_{1}{x}_{1}+...+{t}_{k}{x}_{k}$ where ${t}_{1},...,{t}_{k}\in \mathbb{Z}$

So, I have deduced that the vectors ${x}_{1},...,{x}_{k}\in {\mathbb{R}}^{n}$ so that ${D}_{n}=Int({x}_{1},...,{x}_{k})$ are any vectors such that ${t}_{1}{x}_{1}+...+{t}_{k}{x}_{k}=({v}_{1},...,{v}_{n})$ where ${v}_{1}+...+{v}_{n}$ are even.

However, I feel my solution is certainly lacking and I was wondering if anyone could help guide me in the right direction.

${D}_{n}=$ $({v}_{1},...,{v}_{n})|{v}_{1},...,{v}_{n}\in \mathbb{Z}$ and ${v}_{1}+...+{v}_{n}$ is even}

Find vectors ${x}_{1},...,{x}_{k}$ such that ${D}_{n}=Int({x}_{1},...,{x}_{k})$

My solution:

Firstly, I know that if ${x}_{1},...,{x}_{k}\in {\mathbb{R}}^{n}$, then we define $Int({x}_{1},...,{x}_{k})={t}_{1}{x}_{1}+...+{t}_{k}{x}_{k}$ where ${t}_{1},...,{t}_{k}\in \mathbb{Z}$

So, I have deduced that the vectors ${x}_{1},...,{x}_{k}\in {\mathbb{R}}^{n}$ so that ${D}_{n}=Int({x}_{1},...,{x}_{k})$ are any vectors such that ${t}_{1}{x}_{1}+...+{t}_{k}{x}_{k}=({v}_{1},...,{v}_{n})$ where ${v}_{1}+...+{v}_{n}$ are even.

However, I feel my solution is certainly lacking and I was wondering if anyone could help guide me in the right direction.

asked 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: $\frac{\sqrt{2}}{5}}(3i+4j)$

I'm confused as to what the mathematical logic behind this, how would you explain why we're doing this process?

I'm confused as to what the mathematical logic behind this, how would you explain why we're doing this process?