Revisión de vectores 3D (8/24-8/25) - Preguntas y soluciones, Apuntes de Matemáticas

¡Descarga Revisión de vectores 3D (8/24-8/25) - Preguntas y soluciones y más Apuntes en PDF de Matemáticas solo en Docsity! 3D Vector Review (8/24-8/25) [68 marks] 1a. [5 marks] The following diagram shows the cuboid (rectangular solid) OABCDEFG, where O is the origin, and , , . (i) Find . (ii) Find . (iii) Show that . = 4iOA − →−− = 3jOC − →−− = 2kOD − →−− OB − →−− OF − →−− = −4i + 3j + 2kAG − →−− [7 marks]1b. Find the obtuse angle between the lines OF and AG. 2a. [1 mark] Let A and B be points such that and . Show that . =OA − →−− ⎛ ⎝⎜ 5 2 1 ⎞ ⎠⎟ =OB − →−− ⎛ ⎝⎜ 6 0 3 ⎞ ⎠⎟ =AB − →−− ⎛ ⎝⎜ 1 −2 2 ⎞ ⎠⎟ 2b. [4 marks]Let C and D be points such that ABCD is a rectangle. Given that , show that .=AD − →−− ⎛ ⎝⎜ 4 p 1 ⎞ ⎠⎟ p = 3 2c. [4 marks]Let C and D be points such that ABCD is a rectangle. Find the coordinates of point C. 2d. [5 marks]Let C and D be points such that ABCD is a rectangle. Find the area of rectangle ABCD. ( ) ( ) 3a. [6 marks] Consider the vectors and . (a) Find (i) ; (ii) . Let , where is the zero vector. (b) Find . a = ( )2 −3 b = ( )1 4 2a + b |2a + b| 2a + b + c = 0 0 c 3b. [4 marks]Find (i) ; (ii) . 2a + b |2a + b| [2 marks]3c. Let , where is the zero vector. Find . 2a + b + c = 0 0 c 4a. [3 marks] Consider the points A( , , ) , B( , , ) , and C( , , ) , . Find (i) ; (ii) . 5 2 1 6 5 3 7 6 a + 1 a ∈ R AB − →−− AC − →−− [4 marks]4b. Let be the angle between and . Find the value of for which . q AB − →−− AC − →−− a q = π 2 4c. [8 marks]i. Show that . ii. Hence, find the value of a for which . cosq = 2a+14 14 +280a2√ q = 1.2 [4 marks]4d. Hence, find the value of a for which .q = 1.2