Full answer: 18-31 lead to the main facts about lengths and angles in triangles

Calculate the dot products u v and u wand u (v + w) and w v:

Compute the lengths "u" and "v II and II w II of those vectors. Check the Schwarz inequalities lu. vi < lIullllvll and Iv wi < Ilvllllwll

Find unit vectors in the directions of v and w in Problem 1, and the cosine of the angle e. Choose vectors a, b, c that make 0, 90, and 1800 angles with w

For any unit vectors v and w, find the dot products (actual numbers) of (a) v and-v (b) v + wand v - w (c) v - 2w and v + 2w

Find unit vectors Ul and U2 in the directions of v = (3,1) and w = (2,1,2). Find unit vectors Uland U 2 that are perpendicular to Ul and U2

(a) Describe every vector w = (WI, W2) that is perpendicular to v = (2, -1), (b) The vectors that are perpendicular to V = (1, 1, 1) lie on a __ (c) The vectors that are perpendicular to (1, 1, 1) and (1,2,3) lie on a _

Find the angle e (from its cosine) between these pairs of vectors: (a) v = [~] (c) v = [~] and and w = [~] W= [l] (b) v = UJ and w = HJ (d) v = [~] and w = [ ;}

True or false (give a reason if true or a counterexample if false): (a) If u is perpendicular (in three dimensions) to v and w, those vectors v and w are parallel. " (b) If u is perpendicular to v and w, then u is perpendicular to v + 2 w, (c) If u and v are perpendicular unit vectors then II u - v" = ,.,fi,

The slopes of the arrows from (0,0) to (VI, V2) and (WI, W2) are V2/VI and W2/WI' Suppose the product V2W2/VI WI of those slopes is -1. Show that v . w = 0 and the vectors are perpendicular.

Draw arrows from (0,0) to the points v = (1,2) and w = (-2,1). Multiply their slopes. That answer is a signal that v w = 0 and the arrows are _

If v w is negative, what does this say about the angle between v and w? Draw a 3-dimensional vector v (an arrow), and show where to find all w's with v . w < O.

With v = (1, 1) and w = (1, 5) choose a number c so that w - cv is perpendicular to v. Then find the formula that gives this number c for any nonzero v and w. (Note: cv is the "projection" of w onto v.)

Find two vectors v and w that are perpendicular to (1,0, I) and to each other.

Find nonzero vectors u, v, w that are perpendicular to (1, 1, 1, 1) and to each other.

The geometric mean of x = 2 and y = 8 is .JXY = 4. The arithmetic mean is larger: !(x + y) = . This would come in Example 6 from the Schwarz inequality for v = (.J2, .JS) and w = (,J8, .J2). Find cos e for this v and w.

How long is the vector v = (I, 1, . . ., 1) in 9 dimensions? Find a unit vector u in the same direction as v and a unit vector w that is perpendicular to v.

What are the cosines of the angles a, {J, e between the vector (1,0, -I) and the unit vectors i , j , k along the axes? Check the formula cos2 a + cos2 f3 + cos2 e = 1.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Problems 18-31 lead to the main facts about lengths and angles in triangles.

Using v = randn(3, I) in MATLAB, create a random unit vector u = vl\lvll. Using V = randn(3, 30) create 30 more random unit vectors D.i, What is the average size of the dot products I u . D.i I? In cal cui us, the average f: I cos 8 I d 8 1]'( = 2/]'(.