Please Show Work. One way of describing a plane is by giving a point P in the pl

Question

Please Show Work.

One way of describing a plane is by giving a point P in the plane and two directions v and w in the plane. From this data we can describe the plane as the points you can get by adding to P every possible weighted sum of v and w, i.e. av + bw for various scalars a and b. This description is much like our parametric description of a line, except we use two directions instead of one. [4 points] We have seen that you get the same line if you rescale the direction vector by a nonzero scalar. What is the analogue for planes? That is, if you have two vectors v and w describing a plane, which other pairs of direction vectors will give the same plane? [2 points] Explain why the osculating plane is the plane determined by the velocity and acceleration vectors. [4 points] We call a set of vectors linearly dependent when we can write one vector in the set as a weighted sum of the others. For example, {(1,0,0), (0,1,0), (1,2,0)} is linearly dependent since (1,2,0) = (1,0,0) + 2(0,1,0). So while it's easy to show that a set is linearly dependent, it's trickier to show that it isn't. A set which is not linearly dependent is called linearly independent. To show that a set {u, v, w} is linearly independent, we must show that the only way to have an equation au + bv + cw = 0 is if all the scalars a,b,c are 0. This is because if a is nonzero, then we'd have u = b/a v + c/a w, so the set would be linearly dependent. So here's the problem: If two 3-dimensional vectors v and w are orthogonal, show they must also be linearly independent. (Hint: We have to show that if av + bw = 0), then a and b are both 0. We can do this using the dot product.)

Explanation / Answer

1) a) Any pair of linear combinations of the two, as long as they aren't used in the same ratio, will define the same plane. That is, given v and w, any two vectors of the form (av+bw) and (cv+dw) will give the same plane, assuming they aren't on the same line, which only happens when a/c = b/d. b) Both the velocity and acceleration vectors lie in this plane since the acceleration vector is tangent to the motion. Therefore, by part (a), these two vectors determine the plane (since we can determine the plane by any two vectors that aren't linearly dependent). 2) Suppose that av + bw = 0. Then take the dot product with w. We get a(v . w) + b(w . w) = 0. Since v and w are orthogonal, so their dot product is zero. The dot product of w with itself, however, is non-zero. This implies that b is zero. Similarly, by taking the dot product with v, we get that a is zero. Therefore, a and b are both zero, so the vectors are linearly independent.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.