a-show-that-the-line-through-vectors-mathbfp-and-mathbfq-in-mathbbrn-may-be-written-in-the-parametri

Question

a. Show that the line through vectors $\mathbf{p}$ and $\mathbf{q}$ in $\mathbb{R}^{n}$ may be written in the parametric form $\mathbf{x}=(1-t) \mathbf{p}+t \mathbf{q} .$ (Refer to the figure with Exercises 21 and 22 in Section $1.5 . )$
b. The line segment from $\mathbf{p}$ to $\mathbf{q}$ is the set of points of the form $(1-t) \mathbf{p}+t \mathbf{q}$ for $0 \leq t \leq 1$ (as shown in the figure below). Show that a linear transformation $T$ maps this line segment onto a line segment or onto a single point.

Priyanka Sadarangani · Accepted Answer

Okay, This question There are two parts A and B. So first we will solve a So the question is show that show that the line true vectors B and Q in are in. Maybe the return is maybe written is X equals two when my Nestea P plus take you Okay, so we know that it is only question is X equals toe p plus T Q minus B. And this can be written as p minus tp plus take you. And now we will take a peek. Woman then one dynasty be glass take you So this equation and the sufficient are same. So we have saw that. And now for be barred for be Bart. The ocean is the line segment The line segment segment from P to Q. Is the set off point off the form one minus D p plus? Did you four values off t will be theirs and it was 21 and great 20 questo zero and and we have to show that the linear transformation TV maps this line segment on the way line segment or onto a single point. Okay. For this we will take We will take X equals toe one minus D b plus Take you on t x Will be be when my next DP plus dick. You okay on? We can further sold at one minus. T dp plus de dick. You okay for the values off T this, Okay. And now we will further soul there. Okay, so this will be one way, and this will be the value off the X. Okay, so this has to answers. Okay. And the first one is first one is t X equals to one minus D d p plus t. Did you? So this is the proof that mhm the team maps this line segment on to a line segment. Okay. And if p equals took you here, then this will be DX equals toe one. My nasty d b plus de tp. Okay, then deep in the teepee, cancel on the final will be TV. And this is the This is map This maps on the single point only. Okay, so these two are the answers shall be over. Final answer. Thank you.

Jeffrey Russell · Answer

we&#x27;re going to show that the Parametric Equation X is equal. The x one plus x two minus That&#x27;s won t and why equals y one plus Why? To minus Why won t For tea between your own one. We&#x27;re going to show that this parameter rises the line segments from X one y 12 That&#x27;s two y two. So first off, what&#x27;s solve for T in both equations in the equation for X we get t is equal to x minus X one over x two minus X one and in the equation for why we get t is equal to wide minus y one over y tu minus y one setting both of these equal to each other. We then get X minus X one over x two minus X one is equal to why minus y one over. Why? To minus y one. Okay, now we can rearrange a little bit. Remember that the slope of a line between the points x one y one and x two y two is equal toe y tu minus y one over x two minus X one. In other words, it&#x27;s rise over run Well, we have y tu minus one and x two minus X one over here. So if we multiply by Y tu minus why one on both sides? Then we get why to minus y one over x To my that&#x27;s s one times x minus. X one is equal to y minus y one. And now if we substitute M for this, that is m times X minus x one. It&#x27;s equal to blind minus wide one. This is one of the forms of the equation for a straight line between the points x one, y one and x two y two. But over here we see the t Islamic being from 0 to 1. So when t zero we get X is equal to x one and why is he called a Y one? Minty is one we get X is equal to x one plus x two minus x one or just x two. Similarly, why will be equal to y two? So that means the start and end points of this parameter ization is x one y one and x two y two. So that means that only the part of this line from x one y one x two y two is parameter ice for this privatisation. So that means that we&#x27;ve shown this parameter ization is for a line segment from X one y one x two y two. Now let&#x27;s take these two points negative to seven and three negative one and calculate the parameter ization of the line segment between these two. So substituting in for these two equations we get X is equal to X one, which is negative. Two plus X two is three minus negative, too. Times T, which is equal to negative two plus three minus ing in two or three plus two, is five t now for why we get why is equal to why one which is seven plus Why to which is negative one minus why 17 Currency, which is equal to seven plus negative one minus seven or negative eight. So seven minus eight t So then a parameter ization for the line segment from negative to 7 to 3. Negative one is X equals negative two plus five t and why equals seven minus 18

Wen Zheng · Answer

but a showed that the pair might tricky Questions actually go to x one plus x two minus x one times T Why is it called y one plus y two minus y Juan Castillo? When Tears Between zero and one describes a live segment that joins the points you want pee too, for parts first that we can find it. The question off the line halfway through it. The point he won and pee, too. Direction Wachter of the line It is x two minus X Juan. Why Two? Linus What I want. So we&#x27;re half the question of on the line is ex client is equal to ex Juan? Why one plus he&#x27;d hands x two months x one wide to minus y Want not decide what she is equal to. Zero. Why, it&#x27;s the call to x one y one. When he is going to want ex lie, it&#x27;s equal to Act two. Why two so behalf? But she is between there a lot. So this equation this crabs the line segment that joins the points you want on P two I have to be find the Parametric equations is to represent the line psych might from negative to someone. Two, three, two, one from a half x is equal to you&#x27;re real attacks one by one. It&#x27;s the call to connected to someone. Act too wide, too. It&#x27;s the culture green. Negative one half X is equal to negative, too. US off comes, too. Why is they told you? Someone minus eight. Castillo. This is a monetary question two that represents the life segment from negative to seven two. Three point three two one.

Wen Zheng · Answer

the problem is find a Wachter equation on the Parametric equations was a line segment that vice he took you. He appears and saying Q is new. VW. There&#x27;s a doctor equation This aren&#x27;t you is the cultural one month steam, much parts water. You see, I asked you cams Good luck to you. He covered. Your key is greater than zero on DH smallest. Why the corresponding Parametric equations are actually cool too A Cam&#x27;s mom Honesty Plus you, Counsel, this isn&#x27;t true. You mind pain, Hands to clasp. Why is it good to eat times one dynasty us This is equal to minus house to casting, See is equal to see Tams one Manistee Last W. Thompson, this is Echo two W Manus C comes to us See Andi tease returns and cereal on Smaller than what?

Carson Merrill · Answer

So what we want to do to find the vector equation of a line segment joining two points are not in our one is are vector position Vector is equal to one minus t times. Our first are vector are not plus t times are one are not in Our one are going to be based off of our points. So we know that are not equals One negative 12 negative two And our one is going to be equal to negative 35 and one. So now we can plug in these components and we get that are not Our vector is equal to one minus t times negative 1 to 2 Victor plus t times negative 351 vector Then we multiply us out Combine like terms, um, and make sure we keep all the components together X, y and z we get that are vector the position vector as a function of T is equal to negative one minus two t two plus three t negative two plus three t where t is greater than or equal to zero but less than equal to one. And then we want to show the parametric equations that&#x27;ll be negative on minus two. T for X Y is equal to two plus three. T n Z is equal to negative two plus three t and again this still applies.

Manish Kumar ( Iit K ) · Answer

It is well in this book that we need to find the Brooklyn the cartage. Any question of the line passing through this point? So we do that when the better represent our is given by If R is called to a plus lambda V. Okay over here, landlords question to me is the better part to our and art is just a police encounter along the line. Right. And is the point lying on the tropic? So let&#x27;s say it&#x27;s point. That means you too. I have zero J cap and you&#x27;re okay. Right. So you can write it A J R DR equals to our that it means that cycle policy by the capitalists just take a breath. He calls to a capsule. We are aiming at this is to cap plus, you know your cap plus you&#x27;re okay. Right. And plus of lambda times of Blm to be equal protector, which is parallel to this. These two points of these two sectors is only six months of two for a cap. two minutes of you fuji cap manage to 102 So you can arrange this military cousin R equals two. Two. I can plus learned that times of for I Skyped plus to cap minus two K. Cup. Right? We can also further experience in this little X plus y plus just kick up and I in terms of this one and this one. Okay so we knew the Australian equation in the record form and now in the cartage inform that we know that that would be the form of x minus of excellent upon X two minus X one equal to y minus Y. One upon y tu minus y one equals two. Gentlemen. Gentlemen upon to minus gentlemen, so we know that X one equals to two and two x -21 six minus two. That will be for Equal to Y- of zero upon 2 minutes of people do. Gentlemen, you upon -2 of 0 -2. This will be the cartoons elocution. OK thank you

Problem

Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $…

Problem 27 Easy Difficulty

Answer

A. $\boldsymbol{x}=(1-t) \boldsymbol{p}+t \boldsymbol{q}$
B. See Solution

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Linear Algebra and Its Applications

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A. $\boldsymbol{x}=(1-t) \boldsymbol{p}+t \boldsymbol{q}$B. See Solution

Linear Algebra and Its Applications

Grace H.

Catherine R.

Alayna H.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Catherine R.

Alayna H.

Michael J.

A. $\boldsymbol{x}=(1-t) \boldsymbol{p}+t \boldsymbol{q}$
B. See Solution

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications