- Symbolic functions must be entered using notation similar to the examples given below. - Do NOT include the name of the function or an equal sign. {i.e., do NOT include anything like '1' =' in your answer.} - Decimal numbers cannot be a part of any symbolic function. - Integer numbers [positiveJr negative, and D) and fractions are allowed. - Below are some examples. Example Function oft How to Enter the Answer 3V? tilx'BJ Problem #1: Let P be the point (-3, -2, -1). Suppose that the point Po(-4, -2. 4) is = of the way from P to Q. Find the point Q- Problem #1: Enter the point Q(a,b,c) as a,b,c Just Save Submit Problem #1 for Grading Problem #1 |Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #2: Let u, v, and w be vectors in R". Determine which of the following statements are always true. (i) If u| | = 4. |v/ =5. and ju + v| =7, then u . v = 4. (ii) If u| = 2 and |v = 3, then uv| $6. (iii) The expression (v . w) + u is both meaningful and defined. (A) (i) and (iii) only (B) (ii) and (iii) only (C) all of them (D) none of them (E) (i) and (ii) only (F) (iii) only (G) (ii) only (H) (i) only Problem #2: Select v Just Save Submit Problem #2 for Grading Problem #2 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #3: Let u=(-3, -1, 1) and a = (-3, -1, -1). Find the vector component of u orthogonal to a. Problem #3: Enter your answer symbolically, as in these examples enter your answer in the form a,b,c Just Save Submit Problem #3 for Grading Problem #3 |Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark Problem #4: Find the point on the line x + 2y + 3 =0 that is closest to the point (-2, 0). Problem #4: Enter your answer symbolically, as in these examples enter your answer in the form a,bProblem #5: Find the parametric equations of the line in R' that passes through the points (2, -5, -2) and (-3, 4, 7). (A) x = 1 -5t (B) x = 2 - 5t (C) x = 2- 3t (D) x = 2 - 5t (E) x = 2 - 3t (F) x = 2-4t 1 = -5+9t y = -5+9t V = -5+ 8t 1 = -4+9t y = -5+9t y = -5+9t z = -2+8t z = -2+9t z = -2+8t z = -2+9t z = -3+9t z = -2+9t (G) x = 1 -5t (H) x = 1-5t y = -5+ 10t 1 = -5+9t z = -2+9t 1 = -2+8t Problem #5: Select v Just Save Submit Problem #5 for Grading Problem #5 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #6: Find the area of the triangle with vertices (-5, -2, 2): (-6, -5, -2), and (-1, -6, 3). Problem #6: Enter your answer symbolically, as in these examples Just Save Submit Problem #6 for Grading Problem #6 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #7: Which of the following statements are always true for vectors in RS? (i) If u . (v x w) = 4 then w . (u x v) =-4 (ii) (2u + v) x (u - 3v) = -7 (uxv) (iii) If u is orthogonal to v and w then u is also orthogonal to | wl| v + |vl| w (A) (ii) and (iii) only (B) all of them (C) (i) and (ii) only (D) (i) and (iii) only (E) (iii) only (F) (i) only (G) (ii) only (H) none of themProblem #8: Let I be the set of all ordered pairs of real numbers (u1: up) with up > 0. Consider the following addition and scalar multiplication operations on u = (u1: u2) and v = (v1: v2): utv= (uj + v1 - 6, 2uzvz). ku = (ku1. kuz) Use the above operations for the following parts. (a) Compute u + v for u = (-8, 3) and v = (-1, 5). (b) If the set I' satisfies Axiom 4 of a vector space (the existence of a zero vector), what would be the zero vector? (c) If u = (-8, 3), what would be the negative of the vector u referred to in Axiom 5 of a vector space? (Don't forget to use your answer to part (b) here!) Problem #8(a): enter your answer in the form a,b Problem #8(b): Enter your answer symbolically, as in these examples enter your answer in the form a,b Problem #8(c): Enter your answer symbolically, as in these examples enter your answer in the form a,b Just Save Submit Problem #8 for Grading Problem #8 Attempt #1 Attempt #2 Attempt #3 Your Answer:| 8(a 8(b 8(b) 8(b) 8(c) 8(c) 8(c) Your Mark: | 8(a) B(a) 8(a) 8 (b) 8(b) 8 (b) 8(c) 8(c) 1 8(c) Problem #9: Which of the following sets are closed under scalar multiplication? (i) The set of all vectors in R of the form (a, b) where a + 2b = 2. (ii) The set of all 2 x 2 matrices whose trace is equal to 3. (iii) The set of all polynomials in P2 of the form an + a x + a2 x' where the product ap ajaz = 0. (A) (iii) only (B) none of them (C) all of them (D) (i) only (E) (ii) only (F) (i) and (iii) only (G) (i) and (ii) only (H) (ii) and (iii) only Problem #9: Select vProblem #10: Which of the following sets are closed under addition? (i) The set of all vectors in R- of the form (a, b) where b = a. (ii) The set of all 3 x 3 matrices that have the vector [-3 -2 -1]' as an eigenvector. (iii) The set of all polynomials in P2 of the form an + a x + ay x where ao = as. (A) all of them (B) (i) and (ii) only (C) (ii) and (iii) only (D) (i) and (iii) only (E) none of them (F) (ii) only (G) (iii) only (H) (i) only Problem #10: Select v Just Save Submit Problem #10 for Grading Problem #10 Attempt #1 Attempt #2 Attempt # 3 Your Answer: Your Mark: Problem #11: Let vj = (-3, -1, 1) and v2 = (-3, -1, 0). Which of the following vectors are in span (v1, v2}? (i) (-6, -2. 2) (1i) (-9, -3, 1) (iii) (-15, -5, 3) (A) none of them (B) (ii) only (C) (i) only (D) (ii) and (iii) only (E) (i) and (iii) only (F) (i) and (ii) only (G) (iii) only (H) all of them Problem #11: Select v Just Save Submit Problem #11 for Grading Problem #11 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark:Problem #12: Consider the following scantron sample. STUDENT NUMBER NAME . ........Sample...... . ........ ......... .. Correct.. McMaster Ignore this, University Put the date here SHEET. OF SIGNATURE Correct Sample In pen) Leave these blank - EXAMINATION course Put the course name here .. SECTION..... INSTRUCTOR'S NAME ...... ... .. ANSWER SHEET - use all 9 digtr of your Student number, including leading Beror (if any) STUDENT SEAT NUMBER NUMBER ROOM ROW SEAT Ignore thu part MARKING DIRECTIONS Read these- DO 8 8 1 6 1 32 directions EXAMPLES * Use HB black load pencil only. WRONG 104000 * Do not use ink or bailpoint pens. WRONG Make heavy black marios that fill the circle WRONG Erase clearly any answer you wish to change Make no singy marks on the answer sheet WER SHEET Fill in 9 of these bubbles (one -filled lovable per Column) CLASSROOM AN Put the version.U.U.URUBURURURU Side number here ( filkin one of the bubbles .