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- Question : 1E - (a) v1 = (3, 6) (b) v2 = ( -4, - 8) (d) v4 = (0, 0, -3) (c) v3 = (3, 3, 0)
- Question : 2E - (a) v 1 = (-1, 2) (c) v3 = (1 , 2, 3) (b) v2 = (3 , 4) (d) v4 = (- 1,6, 1)
- Question : 3E - (a) 2u (b) u + v (c) 2u + 2v (d) u - v (e) u + 2v
- Question : 4E - (a) -u+v (b) 3u + 2v (c) 2u + 5v (d) -2u - v (e) 2u- 3v
- Question : 5E - (a)p1(3,5),p2(2,8)(b) P1(5 , - 2, 1), P2(2, 4, 2)
- Question : 6E - (a) PJ(- 6,2), ? 2(-4, - 1) (b) P1(0, 0,0),P2 (- 1,6,1)
- Question : 7E - (a) Find the terminal point of the vector that is equivalent to u = (1, 2) and whose initial point is A(l, 1). (b) Find the initial point of the vector that is equivalent to u = (1 , 1, 3) and whose terminal point is B(-1 , - 1, 2).
- Question : 8E - (a) Find the initial point of the vector that is equivalent to u = (1 , 2) and whose terminal point is B (2, 0) . (b) Find the terminal point of the vector that is equivalent to u = (1 , 1, 3) and whose initial point is A(O, 2, 0).
- Question : 9E - Let u = ( -3, 1, 2, 4, 4), v = (4, 0, -8, 1, 2) , and w = (6, - 1, -4, 3, - 5). Find the components of (a) v-w (b) 6u+2v (c) (2u - 7w) - (8v + u)
- Question : 10E - Let u = (1 , 2, -3, 5, 0) , v = (0, 4, -1 , 1, 2), and w = (7, 1, - 4, - 2, 3). Find the components of (a) v + w (b) 3(2u - v) (c) (3u - v) - (2u + 4w)
- Question : 11E - Let u, v, and w be the vectors in Exercise 11. Find the components of the vector x that satisfies the equation 2u- v + x = 7x + w.
- Question : 12E - Let u , v, and w be the vectors in Exercise 12. Find the components of the vector x that satisfies the equation 3u + v - 2w = 3x + 2w.
- Question : 13E - Which of the following vectors in R6 are parallel to u = (-2, 1, 0, 3, 5, 1)? (a) (4, 2, 0, 6, 10, 2) (b) (4, -2, 0, -6, - 10, - 2) (c) (0, 0, 0, 0, 0, 0)
- Question : 14E - For what value(s) oft, if any, is the given vector parallel to u = (4, -1)? (a) (8t, - 2) (b) (8t, 2t) (c) (1 , t 2 )
- Question : 15E - In each part, sketch the vector u + v + w, and express it in component form.
- Question : 16E - In each part of Exercise 17, sketch the vector u - v + w, and express it in component form.
- Question : 17E - Let u = (1 , -1, 3, 5) and v = (2, 1, 0, -3). Find scalars a and b so that au+ bv = (1, - 4, 9, 18).
- Question : 18E - Let u=(2, 1,0,1, -1) and v = (-2,3,1,0,2) . Find scalars a and b so that au+ bv = ( -8, 8, 3, - 1, 7).
- Question : 19E - Draw three parallelograms that have points A = (0, 0), B = (-1 , 3), and C = (1, 2) as vertices.
- Question : 20E - Verify that one of the parallelograms in Exercise 21 has the ---* ---+ terminal point of AB + AC as the fourth vertex, and then express the fourth vertex in each of the other parallelograms ---* ---+ in terms of AB and A C.
- Question : 21E - 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 1 0 1 1 0 0 0 0 28. 0 0 0 0 0 0 0 0 0 0
- Question : 22E - 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 1 0 1 1 0 0 0 0 28. 0 0 0 0 0 0 0 0 0 0 0 0 0 l 0 0 0 0 0 0 0 0 0 0
- Question : 23E - Give some physical examples of quantities that might be described by vectors in R4 .
- Question : 24E - Is time a vector or a scalar? Write a paragraph to explain your answer.
- Question : 25E - If the sum of three vectors in R3 is zero, must they lie in the same plane? Explain.
- Question : 26E - A monk walks from a monastery gate to the top of a mountain to pray and returns to the monastery gate the next day. What is the monk's displacement? What is the relationship between the monk's displacement going from the monastery gate to the top of the mountain and the displacement going from the top of the mountain back to the gate?
- Question : 27E - figure. (a) What is the sum of the six radial vectors that run from the center to the vertices?(b) How is the sum affected if each radial vector is multiplied by
- Question : 28E - What is the sum of all radial vectors of a regular n-sided polygon? (See Exercise D5.)
- Question : 29E - Consider a clock with vectors drawn from the center to each hour as shown in the accompanying figure. (a) What is the sum of the 12 vectors that result if the vector terminating at 12 is doubled in length and the other vectors are left alone? (b) What is the sum of the 12 vectors that result if the vectors terminating at 3 and 9 are each tripled and the others are left alone? (c) What is the sum of the 9 vectors that remain if the vectors terminating at 5, 11, and 8 are removed?
- Question : 30E - Draw a picture that shows four nonzero vectors in the plane, one of which is the sum of the other three.
- Question : 31E - Indicate whether the statement is true (T) or false (F). Justify your answer. (a) If x + y = x + z, then y = z. (b) Ifu + v = 0, then au+bv = Oforall a and b. (c) Parallel vectors with the same length are equal. (d) If ax= 0, then either a = 0 or x = 0. (e) If au+ bv = 0, then u and v are parallel vectors. (f) The vectors u = ( ../2, .J3) and v = ( Jz, ~ .J3) are equivalent.
- Question : 32E - Prove part (e) of Theorem 1.1.5.
- Question : 33E - Prove part (f) of Theorem 1.1.5.
- Question : 34E - Prove Theorem 1.1.6 without using components.
- Question : 35E - (Numbers and numerical operations) Read how to enter - integers, fractions, decimals, and irrational numbers such as ;r and ../2. Check your understanding of the procedures by converting 7r, ../2, and 1/ 3 to decimal form with various numbers of decimal places in the display. Read about the procedures for performing the operations of addition, subtraction, multiplication, division, raising numbers to powers, and extraction of roots. Experiment with numbers of your own choosing until you feel you have mastered the techniques.
- Question : 36E - (Drawing vectors) Read how to draw line segments in twoor three-dimensional space, and draw some line segments with initial and terminal points of your choice. If your utility allows you to create arrowheads, then you can make your line segments look like geometric vectors.
- Question : 37E - (Operations on vectors) Read how to enter vectors and how - to calculate their sums, differences, and scalar multiples. Check your understanding of these operations by performing the calculations in Example 4.
- Question : 38E - Use your technology utility to compute the components of u = (7 .1 , -3)- 5(../2, 6) + 3(0, ;r) to five decimal places.
- Question : 39E - (a) v = (4, - 3) (b) v = (2, 2, 2) (c) v = (1, 0, 2, 1, 3)
- Question : 40E - (a) v = (- 5, 12) (b) v = (1, -1 , 2) (c) v = (- 2, 3, 3, - 1)
- Question : 41E - (a) llu +vii (c) ll-2u + 2vll (b) !lull+ llvll (d) li3u- 5v + wll
- Question : 42E - (a) llu + v + wll (c) 113vll- 311vll (b) llu - vii (d) !lull - llvll
- Question : 43E - (a) 113u- 5v + wll (c) 11 - llullvll (b) li3uil- 511vll +!lull
- Question : 44E - (a) !lull- 211vll- 311wll (c) lliiu - vliwll(b) !lull + ll - 2vll + ll-3wll
- Question : 45E - Let v = ( -2, 3, 0, 6) . Find all scalars k such that llkvll = 5.
- Question : 46E - Let v = (1, 1, 2, - 3, 1). Find all scalars k such that llkvll =4.
- Question : 47E - (a) u = (3 , 1, 4), v = (2, 2, -4) (b) u = (1, 1, 4, 6), v = (2, -2, 3, -2)
- Question : 48E - (a) u=(l,l, - 2,3),v=(-1,0,5,1) (b) u = (2, - 1, 1, 0, - 2), v = (1, 2, 2, 2, 1)
- Question : 49E - (a) u = (3, 3, 3), v = (1, 0, 4) (b) u=(0,-2, - 1,1),v=(-3,2,4,4) (c) u = (3, -3, -2, 0, -3, 13, 5), v = (- 4, 1, - 1, 5, 0, - 11, 4)
- Question : 50E - (a) u = (1, 2, - 3, 0), v = (5, 1, 2, -2) (b) u = (2, - 1, - 4, 1, 0, 6, - 3, 1), v = (-2, -1, 0, 3, 7, 2, - 5, 1) (c) u = (0, 1, 1, 1, 2), v = (2, 1, 0, - 1, 3)
- Question : 51E - Find the cosine of the angle between the vectors in each part of Exercise 11, and then state whether the angle is acute, obtuse, or a right angle.
- Question : 52E - Find the cosine of the angle between the vectors in each part of Exercise 12, and then state whether the angle is acute, obtuse, or a right angle.
- Question : 53E - A vector a in the xy-plane has a length of 9 units and points in a direction that is 120
- Question : 54E - A vector a in the xy-plane points in a direction that is 47
- Question : 55E - Solve the equation 5x - 2v = 2(w- 5x) for x, given that v = (1, 2, -4, 0) and w = (- 3, 5, 1, 1).
- Question : 56E - Solve the equation 5x- llvllv = llwll (w - 5x) for x with v and w being the vectors in Exercise 17.
- Question : 57E - (a) u
- Question : 58E - (a) !lull
- Question : 59E - (a) u = (3, 2), v = (4, -1) (b) u = (-3, 1, 0), v = (2, -1, 3) (c) u = (0, 2, 2, 1), v = (1, 1, 1, 1
- Question : 60E - (a) u = (4, 1, 1), v = (1, 2, 3) (b) u = (1, 2, 1, 2, 3), v = (0, 1, 1, 5, -2) (c) u = (1, 3, 5, 2, 0, 1), v = (0, 2, 4, 1, 3, 5)
- Question : 61E - Vt= 2'2'2' 2 , Vz = 2'-6'6'6' ( I I I 5) (I I 5 I) V3= 2'6'6, - 6 ,V4= 2'6'-6'6
- Question : 62E - v1 = (- Jz. )6. ~), v2 = (0,- ~
- Question : 63E - Find two unit vectors that are orthogonal to the nonzero vector u =(a, b).
- Question : 64E - For what values of k, if any, are u and v orthogonal? (a) u = (2, k, k), v = (1, 7, k) (b) u = (k, k, 1), v = (k, 5, 6)
- Question : 65E - For which values of k, if any, are u and v orthogonal? (a) u = (k, 1, 3), v = (1, 7, k) (b) u = ( - 2, k, k), v = (k, 5, k)
- Question : 66E - Use vectors to find the cosines of the interior angles of the triangle with vertices A(O, - 1), B(1, - 2), and C(4, 1).
- Question : 67E - Use vectors to show that A(3, 0, 2), B(4, 3, 0), and C (8, 1, -1) are vertices of a right triangle. At which vertex is the right angle?
- Question : 68E - In each part determine whether the given number is a valid ISBN by computing its check digit. (a) 1-56592-170-7 (b) 0-471 -05333-5
- Question : 69E - In each part determine whether the given number is a valid ISBN by computing its check digit. (a) 0-471-06368-1 (b) 0-13-947752-3
- Question : 70E - (Sigma notation) In each part, write the sum in sigma notation. (a) a1b1 + azbz + a3b3 + a4b4 (b) ci + c~ + c~ + c~ + c~ (c) b3 + b4 +
- Question : 71E - (Sigma notation) Write Formula (11) in sigma notation.
- Question : 72E - (Sigma notation) In each part, evaluate the sum for C1 = 3, Cz = -1, c3 = 5, C4 = -6, c5 = 4 d1 = 6, dz = 0, d3 = 7, d4 = - 2, d5 = -3 k=l k=2 k=l
- Question : 73E - (Sigma notation) In each part, confirm the statement by writing out the sums on the two sides. " k=i k=l k=l " k=l k=i k=l " (c) L:cak = c Lak k=l k=i
- Question : 74E - Write a paragraph or two that explains some of the similarities and differences between visible space and higherdimensional spaces. Include an explanation of why R" is referred to as Euclidean space.
- Question : 75E - What can you say about k and v if llkvll = kllvll?
- Question : 76E - (a) The set of all vectors in R2 that are orthogonal to a nonzero vector is what kind of geometric object? (b) The set of all vectors in R3 that are orthogonal to a nonzero vector is what kind of geometric object? (c) The set of all vectors in R2 that are orthogonal to two noncollinear vectors is what kind of geometric object? (d) The set of all vectors in R3 that are orthogonal to two noncollinear vectors is what kind of geometric object?
- Question : 77E - Show that v1 = (t, t. t) and v2 = (t. t
- Question : 78E - Something is wrong with one of the following expressions. Which one is it, and what is wrong? u
- Question : 79E - Let x = (x, y) and x0 = (x0 , y0). Write down an equality or inequality involving norms that describes (a) the circle of radius 1 centered at x0 ; (b) the set of points inside the circle in part (a); (c) the set of points outside the circle in part (a).
- Question : 80E - Ifu and v are orthogonal vectors in R" such that llull = 1 and llvll = 1, then d(u, v) = . Draw a picture to illustrate your result in R2
- Question : 81E - In each part, find llull for n = 5, 10, and 100. (a) u = (1 , ,J2, ,J3, ... , y'n) (b) u=(l,2, 3, ... ,n) [Hint: There exist formulas for the sum of the first n positive integers and the sum of the squares of the first n positive integers. If you don't know those formulas, look them up.]
- Question : 82E - Indicate whether the statement is true (T) or false (F). Justify your answer. (a) If llu + vll2 = llull 2 + llvll 2, then u and v are orthogonal. (b) If u is orthogonal to v and w, then u is orthogonal to v+w. (c) If u is orthogonal to v + w, then u is orthogonal to v and w. (d) If a
- Question : 83E - Indicate whether the statement is true (T) or false (F). Justify your answer. (a) If ku = 0, then either k = 0 or u = 0. (b) If two vectors u and v in R2 are orthogonal to a nonzero vector w in R2 , then u and v are scalar multiples of one another. (c) There is a vector u in R3 such that llu- (1, 1, 1)11 -:::, 3 and llu- ( - 1, - 1, - 1)11 -:::, 3. (d) If u is a vector in R3 that is orthogonal to the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1), then u = 0. (e) Ifu
- Question : 84E - Prove that if u 1 , u2 , . .. , " " are pairwise orthogonal vectors in R", then llu1 + u2 +
- Question : 85E - (a) Use the Cauchy- Schwarz inequality to prove that if a1 and a2 are nonnegative numbers, then .;a;a:; -:::, a1 +a2 --2- The expression on the left side is called the geometric mean of a1 and a2 , and the expression on the right side is the familiar arithmetic mean of a and b, so this relationship states that the geometric mean of two numbers cannot exceed the arithmetic mean. [Hint: Consider the vectors u = (y'al, y'al) and v = (y'al, y'al).] (b) Generalize the result in part (a) for n nonnegative numbers.
- Question : 86E - Use the Cauchy-Schwarz inequality to prove that (a1b1 + a2b2 +
- Question : 87E - (a) Prove the identity u
- Question : 88E - Recall that two non vertical lines in the plane are perpendicular if and only if the product of their slopes is - 1. Prove this using dot products by first showing that if a nonzero vector u =(a, b) is parallel to a line of slope m, then bja = m .
- Question : 89E - Prove Theorem 1.2.4 using Formula (11).
- Question : 90E - (a) Prove part (a) of Theorem 1.2.6. (b) Prove part (b) of Theorem 1.2.6.
- Question : 91E - (a) Use Theorem 1.2.6 to prove part (e) of Theorem 1.2.7 without breaking the vectors into components. (b) Use Theorem 1.2.6 and the fact that 0 = (0)0 to prove part (a) of Theorem 1.2.7 without breaking the vectors into components.
- Question : 92E - As shown in the accompanying figure, let a triangle AXB be inscribed in a circle so tha~e sid~oincides with a diameter. Express the vectors AX and BX in terms of the vectors a and x, and then use a dot product to prove that the angle at X is a right angle.II vii = ~-Determine how to compute dot products and norms with your technology utility and perform the calculations in Examples 1, 2, and 4.
- Question : 93E - (Sigma notation) Determine how to evaluate expressions - involving sigma notation and compute L)3 20 10 em x 15 em x 25 em makes with edges of the box.
- Question : 94E - Use the method of Example 5 to estimate, to the nearest degree, the angles that a diagonal of a box with dimensions 10 20 10 em x 15 em x 25 em makes with edges of the box.
- Question : 95E - (Sigma notation) Let u be the vector in R100 whose ith component is i , and let v be the vector in R100 whose ith component is 1 I (i + 1). Evaluate the dot product u
- Question : 96E - (a) (x,y)=t(2,3) (b) (x, y) = (1, 1) + t(1 , -1)
- Question : 97E - (a) (x , y) = (2, 0) + t(l, 1) (b) (x , y) = t( - 1, -1)
- Question : 98E - (a) (0, 0) and (3, 5) (b) (1, 1, 1) and (0, 0, 0) (c) (1, - 1, 1) and (2, 1, 1)
- Question : 99E - (a) (1, 2) and (- 5, 6) (b) (1, 2, 3) and (-1, - 2, - 3) (c) (1, 2, -4) and (3, - 1, 1)
- Question : 100E - (a) u = (1 , 2); P0 (1, 1) (b) u = (1, -1, 1); P0 (2, 0, 3) (c) u = (3, 2, - 3); P0(0, 0, 0)
- Question : 101E - (a) u = ( -2, 4); P0 (0, 1) (b) u = (5, -2, 1); P0(1, 6, 2) (c) u = (4, 0, -1); P0(4, 0, -1)
- Question : 102E - n = (3, 2, 1); P(-1, - 1, - 1)
- Question : 103E - n = (1 , 1, 4); P(3, 5, -2)
- Question : 104E - (1, 1, 4) , (2, - 3, 1), and (3, 5, -2)
- Question : 105E - (3, 2, 1), (-1, -1 , -1), and (6, 0, 2)
- Question : 106E - (a) Find a vector equation of the line whose parametric equations are X = 2 + 4t, y = - 1 + t , Z = t (b) Find a vector equation of the plane whose parametric equations are (c) Find parametric equations of the plane 3x + 4y- 2z = 4.
- Question : 107E - (a) Find a vector equation of the line whose parametric equations are X= t , y = - 3 + 5t , Z = 1 + t(b) Find a vector equation of the plane whose parametric equations are X = ti + t2, y = 4 + 3ti - t2 , Z = 4ti (c) Find parametric equations of the plane 3x - 5y + z = 32.
- Question : 108E - P(l, 2, 4), Q(1 , - 1, 6), R(l, 4, 8)
- Question : 109E - P(2, 2, 1), Q(O, 3, 4), R(l, - 1, -3)
- Question : 120E - (a) (XI, X2 , X3, X4) = t(l, -2, 5, 7) (b) (xi,X2,X3, X4) = (4, 5,-6, l)+t(l, 1, 1, 1) (c) (XI ,X2, X3, X4)=(- 1,0,4,2)+ti(-3, 5, - 7,4)+ t2(6 , 3, -1 , 2)
- Question : 121E - (a) (XI, X2, X3, X4) = t( -3, 5, -7, 4) (b) (xi, x2, x3, x4) = (5, 6, -5, 2) + t(3, 0, 1, 4) (c) (XI , X2, x3, X4) = fi ( - 4, 7, - 1, 5) + t2(2, 1, - 3, 0)
- Question : 122E - (a) The parametric equations XI = 3t, x2 = 4t, x3 = 7t , x4 = t , x5 = 9t represent a passing through ___ and parallel to the vector ___ _ (b) The parametric equations XI = 3 - 2ti + 5t2 X2 = 4 - 3ti + 6t2 X3 = -2 - 2ti + 7t2 X4 = 1 - 2ti - t2 represent a ___ passing through _ _ _ and parallel to __ _
- Question : 123E - (a) The parametric equations xi = 1 + 2t, x2 = -5 + 3t, x3 = 6t, x4 = - 2 + t, x5 = 4 + 9t represent a _____ passing through and parallel to the vector __ _ (b) The parametric equations XI = 3ti + 5t2 X2 = 4ti + 6t2 X3 = -ti + 5t2 X4 = fi + t2 represent a ___ passing through ___ and parallel to _ _ _
- Question : 124E - Find parametric equations of the plane that is parallel to the plane 3x + 2y - z = 1 and passes through the point P(l, 1, 1).
- Question : 125E - Find parametric equations of the plane through the origin that is parallel to the plane x = ti + t2 , y = 4 + 3ti - t2 , z = 4ti .
- Question : 126E - Which of the following planes, if any, are parallel to the plane 3x + y - 2z = 5? (a) x + y - z = 3 (b) 3x + y - 2z = 0 (c) x +ty - tz =5
- Question : 127E - Which of the following planes, if any, are parallel to the plane x + 2y - 3z = 2? (a) x + 2y - 3z = 3 (b)
- Question : 128E - Find parametric equations of the line that is perpendicular to the plane x + y + z = 0 and passes through the point P(2, 0, 1).
- Question : 129E - Find parametric equations of the line that is perpendicular to the plane x + 2y + 3z = 0 and passes through the origin.
- Question : 130E - Find a vector equation of the plane that passes through the origin and contains the points (5, 4, 3) and (1, - 1, -2).
- Question : 131E - Find a vector equation of the plane that is perpendicular to the x-axis and contains the point P(1, 1, 3).
- Question : 132E - Find parametric equations of the plane that passes through the point P (- 2, 1, 7) and is perpendicular to the line whose parametric equations are x = 4 + 2t , y=-2+3t, z =-5t
- Question : 133E - Find parametric equations of the plane that passes through the origin and contains the line whose parametric equations are X = 2t , y = 1 + t, Z = 2- t
- Question : 134E - Determine whether the line and plane are parallel. (a) x = -5- 4t, y = 1- t, z = 3 + 2t; x + 2y + 3z - 9 = 0 (b) x = 3t, y = 1 + 2t , z = 2- t; 4x + y + 2z = 1
- Question : 135E - Determine whether the line and plane are perpendicular. (a) x = -2- 4t, y = 3- 2t, z = 1 + 2t; 2x + y- z = 5 (b) x = 2 + t , y = 1 - t , z = 5 + 3t; 6x + 6y- 7 = 0
- Question : 136E - Determine whether the planes are perpendicular. (a) 3x - y + z - 4 = 0, x + 2z = -1 (b) x-2y+3z =4,-2x +5y +4z =-1
- Question : 137E - Determine whether the planes are perpendicular. (a) 4x + 3y - z + 1 = 0, 2x - 2y + 2z = - 3 (b) 2x - 3y - z = l , x + 3y- 2z = 12
- Question : 138E - Show that the line x = 0, y = t , z = t (a) lies in the plane 6x + 4y - 4z = 0; (b) is parallel to and below the plane 5x - 3 y + 3z = 1; (c) is parallel to and below the plane 6x + 2y - 2z = -3.
- Question : 139E - Find an equation for the plane whose points are equidistant from (-1 , -4, -2) and (0, -2, 2). [Hint: Choose an arbitrary point (x, y , z) in the plane, and use the distance formula.]
- Question : 140E - (a) 7x - 2y + 3z = - 2 and - 3x + y + 2z + 5 = 0 (b) 2x + 3y - 5z = 0 and 4x + 6y - 10z = 8
- Question : 141E - (a) -3x + 2y + z = -5 and 7x + 3y - 2z = - 2 (b) 5x- 7y + 2z = 0 andy = 0
- Question : 142E - (a) x = 9 - 5t , y = -1- t, z = 3 + t; 2x - 3y + 4z + 7 = 0 (b) X = t, y = t, Z = t; X+ y- 2z = 3
- Question : 143E - (a) x = t , y = t , z = t ; x + y - 2z = 0 (b) x = 3- 4t, y = -2- t, z = 5 + t; 3x- 4y + 5z = 0
- Question : 144E - (a) x = (1 - t)(1, 0) + t(O, 1) (0 .:S: t .:S: 1) (b) x = (1 - t)(1 , 1, 0) + t(O , 0, 1) (0 .:S: t .:S: 1)
- Question : 145E - (a) x = (1 - t)(l, 1) + t(1, - 1) (0 .::: t.::: 1) (b) X= (1 - t)(1 , 1, 1) + t(1 , 1, 0) (0 .:S: t .:S: 1)
- Question : 146E - P( -2, 4, 1), Q(O, 4, 7) 44. P(O, - 6, 5) , Q(3, -1 , 9)
- Question : 147E - Let P = (2, 3, -2) and Q = (7, -4, 1). (a) Find the midpoint of the line segment connecting the points P and Q. (b) Find the point on the line segment connecting P and Q that is ~ of the way from the point P to the point Q.
- Question : 148E - Given that a, b, and care not all zero, find parametric equations for a line in R3 that passes through the point (x0 , y0 , z0) and is perpendicular to the line x = xo +at, y =Yo+ bt, z = zo + ct
- Question : 149E - (a) How can you tell whether the line x = x0 + tv in R3 is parallel to the plane x = x0 + t1v1 + t2v2? (b) Invent a reasonable definition of what it means for a line to be parallel to a plane in R".
- Question : 150E - (a) Letv, w1, and w2 be vectors in R". Show that if vis orthogonal to both w1 and w2 , then vis orthogonal to x = k1 w1 + k2w2 for all scalars k1 and k2 . (b) Give a geometric interpretation of this result in R3 .
- Question : 151E - (a) The equation Ax + By = 0 represents a line through the origin in R2 if A and B are not both zero. What does this equation represent in R3 if you think of it as Ax + By + Oz = 0? Explain.(b) Do you think that the equation Ax1 + Bx2 + Cx3 = 0 represents a plane in R4 if A, B, and Care not all zero? Explain.
- Question : 152E - Indicate whether the statement is true (T) or false (F). Justify your answer. (a) If a, b, and care not all zero, then the line x =at, y = bt , z = ct is perpendicular to the plane ax + by + cz = 0. (b) Two nonparallel lines in R3 must intersect in at least one point. (c) If u, v, and w are vectors in R3 such that u + v + w = 0, then the three vectors lie in some plane through the origin. (d) The equation x =tv represents a line for every vector v in R2
- Question : 153E - (Parametric lines) Many graphing utilities can graph parametric curves. If you have such a utility, then determine how to do this and generate the line x = 5 + 5t , y = - 7t (see Figure 1.3.3).
- Question : 154E - Generate the line L through the point (1, 2) that is parallel to v = ( 1, 1); in the same window, generate the line through the point (1 , 2) that is perpendicular to L. If your lines do not look perpendicular, explain why.
- Question : 155E - Two intersecting planes in 3-space determine two angles of intersection, an acute angle (0 :::: e :::: 90
- Question : 156E - Find the acute angle of intersection between the plane x - y - 3z = 5 and the line X= 2 - t , y = 2t, Z = 3t- 1 (See Exercise T3.)

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