Suppose v,w Rn and c is a scalar. Prove that Span (v + cw,w) = Span (v,w). (See the blue

Given x = (2, 3) and y = (1, 1), calculate the following algebraically and sketch a picture to show the geometric interpretation. a. x + y b. x y c. x + 2y d. 12 x + 12 y e. y x f. 2x y g. _x_ h. x _x_

For each of the following pairs of vectors x and y, compute x + y, x y, and y x. Also, provide sketches. a. x = (1, 1), y = (2, 3) b. x = (2,2), y = (0, 2) c. x = (1, 2,1), y = (2, 2, 2)

Three vertices of a parallelogram are (1, 2, 1), (2, 4, 3), and (3, 1, 5). What are all the possible positions of the fourth vertex? Give your reasoning.\(^4\)

Let A = (1,1,1), B = (1, 1,1), C = (1,1, 1), and D = (1, 1, 1). Check that the four triangles formed by these points are all equilateral.

Let \(\ell\) be the line given parametrically by x = (1, 3) + t (?2, 1), \(t \in \mathbb{R}\). Which of the following points lie on \(\ell\)? Give your reasoning. a. x = (?1, 4) b. x = (7, 0) c. x = (6, 2)

Find a parametric equation of each of the following lines: a. 3x1 + 4x2 = 6 b. the line with slope 1/3 that passes through A = (1, 2) c. the line with slope 2/5 that passes through A = (3, 1) d. the line through A = (2, 1) parallel to x = (1, 4) + t (3, 5) e. the line through A = (2, 1) perpendicular to x = (1, 4) + t (3, 5) f. the line through A = (1, 2, 1) and B = (2, 1, 0) g. the line through A = (1,2, 1) and B = (2, 1,1) h. the line through (1, 1, 0,1) parallel to x = (2 + t, 1 2t, 3t, 4 t)

Suppose x = x0 + tv and y = y0 + sw are two parametric representations of the same line _ in Rn. a. Show that there is a scalar t0 so that y0 = x0 + t0v. b. Show that v and w are parallel.

Decide whether each of the following vectors is a linear combination of u = (1, 0, 1) and v = (-2, 1, 0) . a. x = (1, 0, 0) b. x = (3,-1, 1) c. x = (0, 1, 2)

Let \(\mathcal{P}\) be the plane in \(\mathbb{R}^{3}\) spanned by u = (1, 1, 0) and v = (1, -1, 1) and passing through the point (3, 0, -2). Which of the following points lie on \(\mathcal{P}\)? a. x = (4, -1, -1) c. x = (7, -2, 1) b. x = (1, -1, 1) d. x = (5, 2, 0)

Find a parametric equation of each of the following planes: a. the plane containing the point (-1, 0, 1) and the line x = (1, 1, 1) + t(1, 7, -1) b. the plane parallel to the vector (1, 3, 1) and containing the points (1, 1, 1) and (-2, 1, 2) c. the plane containing the points (1, 1, 2), (2, 3, 4), and (0, -1, 2) d. the plane in \(\mathbb{R}^{4}\) containing the points (1, 1, -1, 2), (2, 3, 0, 1), and (1, 2, 2, 3)

The origin is at the center of a regular m-sided polygon. a. What is the sum of the vectors from the origin to each of the vertices of the polygon? (The case m = 7 is illustrated in Figure 1.17.) Give your reasoning. (Hint: What happens if you rotate the vectors by \(2 \pi / m\)?) b. What is the sum of the vectors from one fixed vertex to each of the remaining vertices? (Hint: You should use an algebraic approach along with your answer to part a.)

Which of the following are parametric equations of the same plane? a. \(\mathcal{P}_{1}:(1,1,0)+s(1,0,1)+t(-2,1,0)\) b. \(\mathcal{P}_{2}:(1,1,1)+s(0,1,2)+t(2,-1,0)\) c. \(\mathcal{P}_{3}:(2,0,0)+s(4,-1,2)+t(0,1,2)\) d. \(\mathcal{P}_{4}:(0,2,1)+s(1,-1,-1)+t(3,-1,1)\)

Given \(\triangle A B C\), let M and N be the midpoints of \(\overline{A B}\) and \(\overline{A C}\), respectively. Prove that \(\overrightarrow{M N}=\frac{1}{2} \overrightarrow{B C}\).

Let ABCD be an arbitrary quadrilateral. Let P, Q, R, and S be the midpoints of AB, BC, CD, and DA, respectively. Use Exercise 13 to prove that PQRS is a parallelogram.

In _ABC, shown in Figure 1.18, _ AD_ = 2 3 _ AB_ and _ CE_ = 25 _ CB_. Let Q denote the midpoint of CD. Show that AQ = c AE for some scalar c, and determine the ratio c = _ AQ_/_ AE_.

Consider parallelogram ABCD. Suppose \(\overrightarrow{A E}=\frac{1}{3} \overrightarrow{A B}\) and \(\overrightarrow{D P}=\frac{3}{4} \overrightarrow{D E}\). Show that P lies on the diagonal \(\overline{A C}\). (See Figure 1.19.)

Given _ABC, suppose that the point D is 3/4 of the way from A to B and that E is the midpoint of BC. Use vector methods to show that the point P that is 4/7 of the way from C to D is the intersection point of CD and AE.

Let A, B, and C be vertices of a triangle in R3. Let x = OA, y = OB, and z = OC. Show that the head of the vector v = 13 (x + y + z) lies on each median of _ABC (and thus is the point of intersection of the three medians). This point is called the centroid of the triangle ABC.

a. Let u, v R2. Describe the vectors x = su + tv, where s + t = 1. What particular subset of such xs is described by s 0? By t 0? By s, t > 0? b. Let u, v,w R3. Describe the vectors x = ru + sv + tw, where r + s + t = 1. What subsets of such xs are described by the conditions r 0? s 0? t 0? r, s, t > 0?

Assume that u and v are parallel vectors in Rn. Prove that Span (u, v) is a line.

Suppose v,w Rn and c is a scalar. Prove that Span (v + cw,w) = Span (v,w). (See the blue box on p. 12.)

Suppose the vectors v and w are both linear combinations of \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}\). a. Prove that for any scalar c, cv is a linear combination of \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}\). b. Prove that v + w is a linear combination of \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}\).

Consider the lines _: x = x0 + tv and m: x = x1 + su. Show that _ and m intersect if and only if x0 x1 lies in Span (u, v).

Suppose x, y Rn are nonparallel vectors. (Recall the definition on p. 3.) a. Prove that if sx + ty =, then s = t = 0. (Hint: Show that neither s _= 0 nor t _= 0 is possible.) b. Prove that if ax + by = cx + dy, then a = c and b = d.

Suppose \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}\) are nonparallel vectors. (Recall the definition on p. 3.) a. Prove that if sx + ty = 0, then s = t = 0. (Hint: Show that neither \(s \neq 0\) nor \(t \neq 0\) is possible.) b. Prove that if ax + by = cx + dy, then a = c and b = d.

Given _ABC, which triangles with vertices on the edges of the original triangle have the same centroid? (See Exercises 18 and 19. At some point, the result of Exercise 25 may be needed, as well.)

Verify algebraically that the following properties of vector arithmetic hold. (Do so for n = 2 if the general case is too intimidating.) Give the geometric interpretation of each property. a. For all x, y Rn, x + y = y + x. b. For all x, y, z Rn, (x + y) + z = x + (y + z). c. =+ x = x for all x Rn. d. For each x Rn, there is a vector x so that x + (x) =

Verify algebraically that the following properties of vector arithmetic hold. (Do so for n = 2 if the general case is too intimidating.) Give the geometric interpretation of each property. a. For all \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}, \mathbf{x}+\mathbf{y}=\mathbf{y}+\mathbf{x}\) b. For all \(\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^{n},(\mathbf{x}+\mathbf{y})+\mathbf{z}=\mathbf{x}+(\mathbf{y}+\mathbf{z})\) c. \(\mathbf{0}+\mathbf{x}=\mathbf{x} \text { for all } \mathbf{x} \in \mathbb{R}^{n}\) d. For each \(\mathbf{x} \in \mathbb{R}^{n}\), there is a vector ?x so that x + (?x) = 0. e. For all c, d \(\in \mathbb{R}\) and \(\mathbf{x} \in \mathbb{R}^{n}\), c(dx) = (cd)x. f. For all c \(\in \mathbb{R}\) and x, y \(\mathbf{x} \in \mathbb{R}^{n}\), c(x + y) = cx + cy. g. For all c, d \(\in \mathbb{R}\) and x \(\mathbf{x} \in \mathbb{R}^{n}\), (c + d)x = cx + dx. h. For all x \(\mathbf{x} \in \mathbb{R}^{n}\), 1x = x.

a. Using only the properties listed in Exercise 28, prove that for any \(\mathbf{x} \in \mathbb{R}^{n}\), we have 0x = 0. (It often surprises students that this is a consequence of the properties in Exercise 28.) b. Using the result of part a, prove that (-1)x = -x. (Be sure that you didn't use this fact in your proof of part a!)