For each of the following pairs of vectors x and y, calculate x y and the angle between

For each of the following pairs of vectors x and y, calculate x y and the angle between the vectors. a. x = (2, 5), y = (5, 2) b. x = (2, 1), y = (1, 1) c. x = (1, 8), y = (7,4) d. x = (1, 4,3), y = (5, 1, 3) e. x = (1,1, 6), y = (5, 3, 2) f. x = (3,4, 5), y = (1, 0, 1) g. x = (1, 1, 1, 1), y = (1,3,1, 5)

For each pair of vectors in Exercise 1, calculate projyx and projxy.

A methane molecule has four hydrogen (H) atoms at the points indicated in Figure 2.7 and a carbon (C) atom at the origin. Find the H C H bond angle. (Because of the result of Exercise 1.1.4, this configuration is called a regular tetrahedron.)

Find the angle between the long diagonal of a cube and a face diagonal.

Find the angle that the long diagonal of a 3 4 5 rectangular box makes with the longest edge.

Suppose \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n},\|\mathbf{x}\|=3,\|\mathbf{y}\|=2\), and the angle \(\theta\) between x and y is \(\theta\) = arccos(?1/6). Show that the vectors x + 2y and x ? y are orthogonal.

Suppose \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n},\|\mathbf{x}\|=\sqrt{2},\|\mathbf{y}\|=1\), and the angle between x and y is \(3 \pi / 4\). Show that the vectors 2x + 3y and x ? y are orthogonal.

Suppose x, y, z \(\in \mathbb{R}^{2}\) are unit vectors satisfying x + y + z = 0. Determine the angles between each pair of vectors.

Let \(\mathbf{e}_{1}=(1,0,0), \mathbf{e}_{2}=(0,1,0)\), and \(\mathbf{e}_{3}=(0,0,1)\) be the so-called standard basis for \(\mathbb{R}^{3}\). Let \(\mathbf{x} \in \mathbb{R}^{3}\) be a nonzero vector. For i = 1, 2, 3, let \(\theta_{i}\) denote the angle between x and \(\mathbf{e}_{i}\). Compute \(\cos ^{2} \theta_{1}+\cos ^{2} \theta_{2}+\cos ^{2} \theta_{3}\).

Let x = (1, 1, 1, . . . , 1) Rn and y = (1, 2, 3, . . . , n) Rn. Let n be the angle between x and y in Rn. Find lim n n. (The formulas 1 + 2+ +n = n(n + 1)/2 and 12 + 22 + +n2 = n(n + 1)(2n + 1)/6 may be useful.) _

Suppose x, v1, . . . , vk Rn and x is orthogonal to each of the vectors v1, . . . , vk. Show that x is orthogonal to any linear combination c1v1 + c2v2 + +ckvk.6

Use vector methods to prove that a parallelogram is a rectangle if and only if its diagonals have the same length.

Use the algebraic properties of the dot product to show that _x + y_2 + _x y_2 = 2 _ _x_2 + _y_2 _ . Interpret the result geometrically.

Use the dot product to prove the law of cosines: As shown in Figure 2.8, c2 = a2 + b2 2ab cos .

Use vector methods to prove that a triangle that is inscribed in a circle and has a diameter as one of its sides must be a right triangle. (Hint: See Figure 2.9. Express the vectors u and v in terms of x and y.)

a. Let y Rn. If x y = 0 for all x Rn, then prove that y = 0b. Suppose y, z Rn and x y = x z for all x Rn. What can you conclude? (Hint: Apply the result of part a.)

If x = (x1, x2) R2, set (x) = (x2, x1). a. Check that (x) is orthogonal to x. (Indeed, (x) is obtained by rotating x an angle /2 counterclockwise.) b. Given x, y R2, show that x (y) = (x) y. Interpret this statement geometrically.

Prove the triangle inequality: For any vectors \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n,\|\mathbf{x}+\mathbf{y}\| \leq\|\mathbf{x}\|+\|\mathbf{y}\|\). (Hint: Use the dot product to calculate \(\|\mathbf{x}+\mathbf{y}\|^2\).)

a. Give an alternative proof of the Cauchy-Schwarz Inequality by minimizing the quadratic function Q(t) = _x ty_2. Note that Q(t) 0 for all t . b. If Q(t0) Q(t) for all t , how is t0y related to x_? What does this say about projyx?

Use the Cauchy-Schwarz inequality to solve the following max/min problem: If the (long) diagonal of a rectangular box has length c, what is the greatest that the sum of the length, width, and height of the box can be? For what shape box does the maximum occur?

a. Let x and y be vectors with _x_ = _y_. Prove that the vector x + y bisects the angle between x and y. (Hint: Because x + y lies in the plane spanned by x and y, one has only to check that the angle between x and x + y equals the angle between y and x + y.) b. More generally, if x and y are arbitrary nonzero vectors, let a = _x_ and b = _y_. Prove that the vector bx + ay bisects the angle between x and y.

Use vector methods to prove that the diagonals of a parallelogram bisect the vertex angles if and only if the parallelogram is a rhombus. (Hint: Use Exercise 21.)

Given_ABC withD onBC, as shown in Figure 2.10, prove that ifAD bisects _ BAC, then _ and y = AC; express AD in two ways as a linear combination of x and y and use Exercise 1.1.25.)

Use vector methods to show that the angle bisectors of a triangle have a common point. (Hint: Given _OAB, let x = OA, y = OB, a = _ OA_, b = _ OB_, and c = _ AB_. If we define the point P by OP = 1 a+b+c (bx + ay), use part b of Exercise 21 to show that P lies on all three angle bisectors.)

Use vector methods to show that the altitudes of a triangle have a common point. Recall that altitudes of a triangle are the lines passing through a vertex and perpendicular to the line through the remaining vertices. (Hint: See Figure 2.11. Let C be the point of intersection of the altitude from B and the altitude from A. Show that \(\overrightarrow{O C}\) is orthogonal to \(\overrightarrow{A B}\).)

Use vector methods to show that the perpendicular bisectors of the sides of a triangle intersect in a point, as follows. Assume the triangle OAB has one vertex at the origin, and let x = OA and y = OB. Let z be the point of intersection of the perpendicular bisectors of OA and OB. Show that z lies on the perpendicular bisector of AB. (Hint: What is the dot product of z 12 (x + y) with x y?)