Download Applied Mathematics Problems: Vector Operations and Linear Algebra and more Exercises Mathematics in PDF only on Docsity!
Mekelle University College of Natural and Computational Sciences Department of Mathematics Applied mathematics-I Work Sheet
1. Find ⃗ a. ⃗ b and ‖ 2 ⃗ a − 4 ⃗ b ‖ if
a) ⃗ a = 3 i + 2 j + 3 k , b ⃗ = i − 4 j + k b) a ⃗ =⟨ −2,6,1⟩ , ⃗ b =⟨ 3 , −3,1 ⟩
- Determine the value of x such that ‖ a ‖=√ 39 , where a =⟨ 1 , x , − 5 ⟩
3. Let v =⟨ −1,5,− 2 ⟩∧ w =⟨ 3, 1, 1 ⟩. Then , find the vector u such that u + v + w = 2 j + k.
4. Given three vectors a^ ,^ b and c such that a + b + c =0.If‖ a ‖=3,‖ b ‖=1, ‖^ c^ ‖ =4,then evaluate a^ ∙^ b + b^ ∙^ c^ + a^ ∙^ c.
- Given that u^.^ u =3,^ u^.^ v =^4 ,and v^.^ v^ =^7 then find the values of a) ( u + v ). ( u − v ) b) ( u + v ). ( u + v )
6. Given two vectors u and v with ‖ u ‖= 2 , ∧‖ v ‖= 1 .If the angle between them is 600 , then find
a).‖ u − 2 v ‖ b). ‖ 2 u − 3 v ‖
- Determine whether the given vectors are orthogonal, parallel, or neither
a) a =⟨ −5,3,7⟩ , b =⟨ 6 , −8,2⟩
b) a =⟨ 4, 6 ⟩ , b =⟨−3,2 ⟩
8. Find a real number β such that the vectors a =⟨ β , −3,1⟩and b =⟨ β , β , 2 ⟩ are perpendicular.
9. If ‖ u ‖= 3 , ‖ v ‖= 5. Determine the value of x for which the vectors u + x v and u − x v will be mutually
perpendicular. 10.Show that if v is orthogonal to both w and u , then v is orthogonal to α w + β u for all scalars α ∧ β.
11.Find a unit vector in the direction of u − 2 v if u =⟨−1,1,0 ⟩ and v =⟨ 1,2,1 ⟩.
12.Find a vector that has the same direction as⟨ −2, 4, 2 ⟩ but has length6.
- Find the angle between the vectors a ⃗ and (^) b ⃗.
a) a = 4 i + 3 j , b = 2 i + j b) a =⟨ 3 , −1,5⟩ , b =⟨−2,4,3⟩
14. If^ ‖ a ‖=4,‖ b ‖=^2 , ‖ c ‖=^6 ∧¿the angle between a^ ,^ b ∧ c^ is 600 .Find the norm of the vectors a + b + c.
15. Find the direction cosines of the vectors a =⟨ 2,1,2 ⟩ and b =⟨ 6,3 , − 2 ⟩
16.If a vector has direction angles α = π 4 and β = π 3 , find the third direction angle γ
- Find the scalar and vector projections of ⃗ a onto (^) b ⃗ if:
a) ⃗ a =⟨ 1 , −4,3 ⟩ , b ⃗ =⟨−2,6,2⟩
b) ⃗ a = 2 i − 3 j + 5 k , (^) b ⃗ = 5 i + 3 j − 2 k
18. If a =⟨ 3,0,1 ⟩ , find a vector b such that compa^ b^ =2.
- Find the area of the triangle determined by the points P ,Q , and R. a) (^) P ( 1 , −1,2) ,Q ( 2,0 , − 1 ) , R ( 0,2,1) b) P ( 1,1,1) , Q ( 2,1,3) , R ( 3 , −1,1)
- Find the volume of the parallelepiped determined by the vectors (^) a ⃗ , b ⃗ ∧⃗ c.
a) ⃗ a =⟨ 1, 2, 3 ⟩ , ⃗ b =⟨−1, 1, 2 ⟩ , ⃗ c =⟨ 2, 1, 4 ⟩
b) (^) ⃗ a = i + j , ⃗ b = j + k , ⃗ c = i + j + k
21. Find ⃗ a × ⃗ b , 2 ⃗ a × 4 ⃗ b , ‖ 2 ⃗ a × 4 b ⃗‖ if
a) (^) ⃗ a = 5 i − 6 j − k , ⃗ b = 3 i +¿k b) (^) ⃗ a = 4 i − 6 j + 2 k , ⃗ b =− 2 i + 3 j – k 22.If (^) ⃗ a =¿ 2,0,− 1 ¿ , ⃗ b =←3,1,0 ¿ , and c ⃗=¿1,−2,4 ¿ , then find (^) a ⃗ × ( b ⃗ × ⃗ c ) and (^) ( a ⃗ × ⃗ b ) × ⃗ c.
- If (^) ⃗ a. ⃗ b =√ 3 and (^) a ⃗ × ⃗ b =⟨ 1,2,2⟩ , then find the angle between a ⃗ and (^) b ⃗.
24.Find two unit vectors that are orthogonal to both⟨ 1,0,1 ⟩and⟨ 0,1,1⟩.
25.Show that (^) ¿∨ a ×b ∨¿^2 +¿∨ a·b ∨¿^2 =¿∨ a ∨¿^2 ¿∨ b ∨¿^2.
- Find the vector, parametric and symmetric equations of the line: a) through the point (1, 5, 2 ) and parallel to the vector 〈 2 , −2,7 〉. b) the point (1,0,6)and perpendicular to the plane x + 3 y + z = 5. c) the point (1,0,6) and perpendicular to both i + j and j + k. d) the point (4,-1,3) and orthogonal to both lines, L 1 :^ x =^1 +^3 t^ ,^ y^ =^2 t , z =− 1 − t ∧¿ L 1 : x = 1 − 2 t , y =− t , z =− 1 + 2 t.
27. Let l 1 be the line through ( 1 , −6, 2 ) with direction vector ⟨ 1, 2, 1 ⟩ and l 2 be the line through (0, 4,1) with
direction vector ⟨ 2,1,2⟩. Determine whether these lines are parallel, intersect or they are skew. If they intersect,
find the point at which they intersect and the angle between them.
- Find the distance D between the point and the line L given by a. (3, -1, 4); L : x =− 2 + 3 t , y =− 2 t , z = 1 + 4 t. b. ( 4, 1 , − 2 ) ; L : x = 1 + t , y = 3 − 2 t , z = 4 − 3 t
- Find the equation of the plane passes: a) through the points P (1, 1,− 1 ) , R (2, 0, 2 )and Q (0, -2, 1). b) through the point P (2, 4, 5 ) and orthogonal to the line with parametric equation L 1 : x = 5 + t , y = 1 + 3 t , z = 4 t.
- Determine whether the lines L 1 and L 2 are parallel, skew, or intersecting. If they intersect, find the point of intersection_._ a) L 1 :^ x − 4 2
y + 5 4
z − 1 − 3
∧ L 2 :
x − 2 1
y + 1 3
z 2 b) L 1 :^ x − 1 2
y 1
z − 1 4
¿ L 2 :
x 1
y + 2 2
z + 2 3
- Find the point of intersection of the lines L 1 :^ x =^1 +^2 t^ ,^ y^ =^2 +^3 t^ ,^ z =^3 +^4 t^ and L 1 : x = 2 + p , y = 4 + 2 p , z =− 1 − 4 p. Then find the plane determined by these two lines.
- Find the inverse of the matrix A=
[
4 − 2 1 7 3 3 2 0 1
]
.
- Consider the matrix B =
[
]
a) Find the determinant of B by first expanding along row 1. b) Find the determinant of B by first expanding along column 2
45. Let A , B and C be 4 × 4 matrices such that | A |=5,| B |=− 3 and | C |=^
. Find the determinant of the matrix 2 A B − 1 C 3 B T
- Given that
a b c d e f
g h i |
= (^5). Find
2 d 2 e 2 f a b c
g − a h − b i − c |
- Prove that
2 2 2
a b c
a b c
( a b )( b c )( c a )
- Let
A
x x
x
x
, where x^ ^ R.
a) Find the value(s) of x such that A is non-singular.
b) If ¿^3 , find A
1 .
46. Given the linear system AX = B , with A =
[
2 − 2 γ
3 − 2 0 ]^
, B =
[
β ]^
where γ ∧ β are two real numbers, find the values of these parameters such that: i. The system is inconsistent; ii. The system has infinitely many solutions; iii. The system has exactly one solution.
47. Consider the matrix A =
[
1 7 5 ]
. i. Use Gaussian elimination to find the determinant of A. ii. Use the cofactor expansion to find the determinant of A.
48. Solve the following systems using Gaussian-Elimination methods.
a) x 1 +^ x 2 +^2 x 3 =^8 b) 2 x 1 +^2 x 2 +^2 x 3 =^0 − x 1 − 2 x 2 + 3 x 3 = 1 − 2 x 1 + 5 x 2 + 2 x 3 = 1 3 x 1 − 7 x 2 + 4 x 3 = 10 8 x 1 + x 2 + 4 x 3 =− 1
49. Solve using Cramer’s rule:
x 1 + x 2 + 2 x 3 = 92 x 1 + 4 x 2 − 3 x 3 = 13 x 1 + 6 x 2 − 5 x 3 = 0
