French-Azerbaijani University (UFAZ) HOMEWORK 2 Vectors. Linear operations on vectors. Dot product (Scalar product). Cross product (Vector product). Mixed product (Scalar triple product). 1) Given the points A(−1, −2, 4) and B(1, 3, −1) −→ (a) Find the components of vector 3BA −→ −−→ (b) Find the components of vector 2OA + OB 2) Find the unit vector that has the same direction as ~v = ~i + 2~j − 2~k . 3) Find the scalar t (or show that there is none) so that the vector ~v = 0.5~i − t~j + 1.5t~k is a unit vector. −→ 4) If A (1; 3; 2) and B (5; 8; −1) then nd out the AB vector and it's length. → − → − → − − 5) Find out magnitude of → a = 4 i + 2 j − 4 k vector. −→ → − −−→ → − → − → − −−→ → − → − 6) Let vectors AB = i + 2 j , BC = −4 i − j , CD = −5 i − 3 j are given in a plane. Prove that ABCD is a trapezoid. → − − 7) Find out dot product of → a and b vectors. → − → − − − (a) |→ a | = 3, | b | = 1, angle between → a and b is 45 . → − → − ◦ → − → − (b) | a | = 6, | b | = 7, angle between a and b is 120 . ◦ → − → − − − 8) Find dot product of → a and b if it's known that they formed a 30 degrees angle and |→ a | = 4, | b | = 5. → − − 9) Find the scalar product of → a and b vectors. → − − → − → → − → − − (a) → a =4 i − j , b =− i −7j → − → − − (b) → a (2; 1), b (1; −3) → − − (c) → a (3; 2; −5), b (10; 1; 2) → − → − → − → − → − − (d) → a = 2 i + j + 5 k , b = 7 i − 9j − k → − − 10) Find the scalar product of → a and b vectors. → − − → − → → − → − − (a) → a =3 i −2j , b = i + j → − → − − → − → → − → − − (c) → a =2i − k + j, b = k − j → − − → − → → − → − − (b) → a =5 i , b =− i −2j → − → − → − → − → − → − − (d) → a = i +2j +3k, b =2j + k → − → − − − − 11) If given that |→ a | = 3, | b | = 5 then for what values of α the vector → a +α b and → a −α b are perpendicular to each other? → − − 12) Find the angle between → a and b vectors. → − → − → − → − → − − − − (a) a (1; 2) , b (2; 4) (b) → a (1; 2) b (−2; 1) (c) → a (1; 0; 0) , b (1; 1; 0) (d) → a (1; −1; 1), b (4; 4; −4) 13) Let ~u and ~v be non-zero vectors such that |~u − ~v | = |~u + ~v |. Show that if, ~u = u1~i + u2~j + u3~k and ~v = v1~i + v2~j + v3~k , then u1 v1 + u2 v2 + u3 v3 = 0. − − − 14) Find the coordinates of the → x the vector that satises the → a ·→ x = 3 condition and collinear to the → − vector a (2; 1; −1). → − → − − → − → − − → − −c | = 4 and → − − − 15) If |→ a | = 3, | b | = 1, |→ a + b +→ c = 0 then nd the sum → a · b + b ·→ c + −c · → a =?. − − − 16) Vector → x is perpendicular to the → a 1 (2; 3; −1) and → a 2 (1; −2; 3) vectors and satises the condition − → − → − → → − − x · (2 i − j + k ) = −6 . Find the coordinates of the → x vector. French-Azerbaijani University (UFAZ) → − → − → − → − → − → − − 17) Find the angle between → a = − i + j and b = i − 2 j + 2 k vectors. 18) Find the sides and inner angles of the triangle with vertices at the points A (−1; −2; 4), B (−4; −2; 0), C (3; −2; 1). 19) A(1, 2, 3), B(−3, 2, 4) and C(1, −4, 3) are vertices of a triangle. Show that the triangle is right-angled and nd its area. 20) Simplify the expression − → − → − → − → − → − → → − → − → − → − → − → − → − → − → − → − → − → − (a) i × j + k + j × i + k + k × i + j (b) i × j − k + j × k − i + k × i − j 21) Simplify the expression. − → − → − → − − → − → → − → − → − → − → j + k − j × i + k + k × i + j + k → − − → → − − → − → − − → − − (b) → a + b +→ c × −c + → a + b +→ c × b + b −→ c ×− a → − → − → − − −c − → − −c × → − (c) 2→ a + b × (→ a)+ b +→ a + b (a) i × → − − 22) Find the vector product (cross product) of → a and b vectors. → − → − → − → − → − → − → − − (a) → a = 4 i + 9 j + k , b = −3 i − 2 j + 5 k → − − → − → − (c) → a (2; 3; 0), b (0; 3; 2) → − − (b) → a (1; −5; −1), b (2; −3; 3) → − − (d) → a =3 i , b =2k → − − (e) → a (1; 1; 0), b (1; −1; 0) → − − 23) Find the vector product (cross product) of → a and b vectors. → − → − → − → − → − → − → − → − → − → − → − → − → − → − − − (a) → a = 3 i − 2 j + 4 k , b = −2 i + j − 3 k (b) → a = −5 i + 5 j − k , b = − i − 5 j + 5 k → − → − → − → − → − → − → − − (c) a (2; 3; 0), b (0; 3; 2) (d) a = 2 k , b = 2 j (e) → a (1; −1; 0), b (0; 1; −1) → − − (f) → a (0; 1; 1), b (0; 0; 1) 24) Find unit vector that is orthogonal to both vectors ~u = −2~i + 4~j + 2~k and ~v = 3~i + 2~j − 3~k . → − → − → − → − → − → − → − → − → − → − − → − → − 25) If → a = 3 i − j + k , b = i + 2 k then for what values of α and β the vector α i + 3 j + β k will → − − be collinear to the vector → a × b. → − − 26) The vectors → a (3; −1; 2) and b (1; 2; −1) are given. Find the following vector product. → − → − → − → − → → − → − → − → − − (a) a × b (b) 2 a + b × b (c) 2 a − b × 2 a + b → − → − → − → − − 27) Calculate the area of the parallelogram where the sides are → a = 2 i +3 j +5 k and b = i +2 j + k . −→ → − → − −−→ → − → − 28) Calculate the area of the parallelogram where the sides are OA = i + j and OB = −3 j + k . → − → − − − 29) The → a and b vectors are formed a 45 degrees angle and |→ a | = | b | = 5. Calculate the area of the → − → − − − triangle where the sides are → a − 2 b and 3→ a +2 b . 30) Calculate the area of the triangle ∆ABC , where A = (1; 1; 1) , B = (2; 3; 4) and C = (4; 3; 2). 31) Calculate the area of the triangle ∆ABC , where A = (1; 2; 3) , B = (3; 2; 1) and C = (1; −1; 1). → − → − − − 32) The → a and b vectors are formed a 30 degrees angle and |→ a | = 6, | b | = 8. Calculate the area of the → − → − → − → − parallelogram where the sides are a − 2 b and 3 a + 2 b . → − → − → → − → − → − → − − → − 33) If a (2; 1; −3) , b (1; −1; 1) then nd the vector a × a + b + a × a × b . French-Azerbaijani University (UFAZ) − 34) Find the mixed product → a · → − → − → − → → − − −c vectors. b × −c of → a , b and → → − → − → − → − → − → − → − → − − −c = 3 i + j − 2 k (a) → a = i + 2 j + 3 k , b = − i − j + 2 k ,→ − → − → − − → − → − → − → → − → − → − → − − (b) → a = 2 i − 3 j + k , b = 5 i − j + 4 k ,→ c =− i + j −3k → − → − → − → − − → − − −c (0; 0; 6) (d) → − (c) → a (1; 2; 3), b (0; 4; 5) , → a = 2 k , b = 2 j ,→ c =2i − 35) Find the mixed product → a · → − → − → − → → − − −c vectors. b × −c of → a , b and → → − → − → − → − → − → − → − → − − −c = 7 i + 8 j + 9 k (a) → a = i + 2 j + 3 k , b = 4 i + 5 j + 6 k ,→ → − − → − → → − → − → − → − − −c = − i + 2 k (b) → a = 2 i − j , b = 2 j − k ,→ → − − −c (2; −1; 8) (c) → a (3; 4; −5), b (8; 7; −2) , → → − → − → − → − → − → − → − − −c = − j − k (d) → a = i + k , b = − i − j ,→ → − − −c (1; 1; 1) vectors are coplanar or not. 36) Determine that → a (1; 5; 2) , b (−1; 1; −1) , → → − − −c vectors are coplanar or not. 37) Check that the following → a , b ,→ − → − → − − → − → − → − → → − → − → − → − − (a) → a = i + j + k , b = −2 i + 3 j − 2 k , → c =3 i −2j +3k → − → − → − − → − → − → − → − → − → − → − − (b) → a =2 i − j +3k, b = i +2j −3k , → c = i +3j −2k 38) Prove that A (5; 7; −2) , B (3; 1; −1) , C (9; 4; −4) , D (1; 5; 0) points are in one plane. 39) Determine whether these points A, B, C, D are in one plane or not. (a) A (1; 0; 1) , B (2; 1; −2) , C (1; 2; 0) , D (−1; 1; −1) (b) A (1; 2; −1) , B (0; 1; 5) , C (−1; 2; 1) , D (2; 1; 3) 40) For what value(s) of m are the following four points on the same plane? A(m, 3, −2), B(3, 4, m), C(2, 0, −2), and D(4, 8, 4) 41) Find the volume of the triangular pyramid with vertices (a) A (1; 1; 1) , B (3; 2; 1), C (5; 3; 2), D (3; 4; 5). (b) A (2; 0; 0), B (0; 3; 0), C (0; 0; 6), D (2; 3; 8).
