MATH 311-504Topics in Applied MathematicsLecture 13:Review for Test 1.Topics for Test 1Vectors (Williamson/Trotter 1.1–1.2, 1.4, 1.6, 2.2C)• Vector addition and scalar multiplication• Length of a vector, angle between vectors• Dot product, orthogonality• Cross product, mixed triple product• Linear dependenceAnalytic geometry (Williamson/Trotter 1.3, 1.5–1.6)• Lines and planes, parametric representation• Equations of a line in R2and of a plane in R3• Distance from a point to a line in R2or from a point to aplane in R3• Area of a triangle and a parallelogram in R3• Volume of a parallelepiped in R3Topics for Test 1Systems of linear equations (Williamson/Trotter 2.1–2.2)• Elimination and back substitution• Elementary operations, Gaussian elimination• Matrix of coefficients and augmented matrix• Elementary row operations• Row echelon form and reduced row echelon form• Free variables, parametric representation of the solution set• Homogeneous systems, checking for linear independence ofvectorsTopics for Test 1Matrix algebra (Williamson/Trotter 2.3–2.4)• Matrix addition and scalar multiplication• Matrix multiplication• Diagonal matrices, identity matrix• Matrix polynomials• Inverse matrixDeterminants (Williamson/Trotter 2.5)• Explicit formulas for 2×2 and 3×3 matrices• Elementary row and column operations• Row and column expansions• Test for linear dependenceSample problems for Test 1Problem 1 (25 pts.) Let Π be the plane in R3passingthrough the points (2, 0, 0), (1, 1, 0), and (−3, 0, 2). Let ℓ bethe line in R3passing through the point (1, 1, 1) in thedirection (2, 2, 2).(i) Find a parametric representation for the line ℓ.(ii) Find a parametric representation for the plane Π.(iii) Find an equation for the plane Π.(iv) Find the point where the line ℓ intersects the plane Π.(v) Find the angle between the line ℓ and the plane Π.(vi) Find the distance from the origin to the plane Π.Problem 2 (15 pts.) Let f (x) = a cos 2x + b cos x + c.Find a, b, and c so that f (0) = 0, f′′(0) = 2, and f′′′′(0) = 10.Sample problems for Test 1Problem 3 (20 pts.) Let A =0 −2 4 12 3 2 01 0 −1 11 0 0 1.Find the inverse matrix A−1.Problem 4 (20 pts.) Evaluate the following determinants:(i)0 −2 4 12 3 2 01 0 −1 11 0 0 1, (ii)2 −2 0 3−5 3 2 11 −1 0 −32 0 0 −1.Bonus Problem 5 (15 pts.) Find the volume of thetetrahedron with vertices at the points a = (1, 0, 0),b = (0, 1, 0), c = (0, 0, 1), and d = (2, 3, 5).Problem 1 Let Π be the plane in R3passingthrough the points (2, 0, 0), (1, 1, 0), and (−3, 0, 2).Let ℓ be the line in R3passing through the point(1, 1, 1) in the direction (2, 2, 2).(i) Find a parametric representation for the line ℓ.Parametric representation: t(2, 2, 2) + (1, 1, 1).The line ℓ passes through the origin (t = −1/2).Hence an equivalent representation is s(2, 2, 2).Problem 1 Let Π be the plane in R3passingthrough the points (2, 0, 0), (1, 1, 0), and (−3, 0, 2).Let ℓ be the line in R3passing through the point(1, 1, 1) in the direction (2, 2, 2).(ii) Find a parametric representation for the plane Π.Since the plane Π contains the points a = (2, 0, 0),b = (1, 1, 0), and c = (−3, 0, 2), the vectorsb −a = (−1, 1, 0) and c − a = (−5, 0, 2) areparallel to Π. Clearly, b − a is not parallel toc − a. Hence we get a parametric representationt1(b − a) + t2(c − a) + a == t1(−1, 1, 0) + t2(−5, 0, 2) + (2, 0, 0).Problem 1 Let Π be the plane in R3passingthrough the points a = (2, 0, 0), b = (1, 1, 0), andc = (−3, 0, 2). Let ℓ be the line in R3passingthrough the point (1, 1, 1) in the direction (2, 2, 2).(iii) Find an equation for the plane Π.Vectors b − a = (−1, 1, 0) and c − a = (−5, 0, 2) are parallelto Π =⇒ their cross product p is orthogonal to Π.p =i j k−1 1 0−5 0 2=1 00 2i −−1 0−5 2j +−1 1−5 0k= 2i + 2j + 5k = (2, 2, 5).A point x = (x, y, z) is in the plane Π if and only ifp · (x − a) = 0 ⇐⇒ 2(x − 2) + 2y + 5z = 0⇐⇒ 2x + 2y + 5z = 4Problem 1 Let Π be the plane in R3passingthrough the points (2, 0, 0), (1, 1, 0), and (−3, 0, 2).Let ℓ be the line in R3passing through the point(1, 1, 1) in the direction (2, 2, 2).(iv) Find the point where the line ℓ intersects theplane Π.Let x0= (x, y, z) be the point of intersection.Then x0= s(2, 2, 2) for some s ∈ R and also2x + 2y + 5z = 4.2(2s) + 2(2s) + 5(2s) = 4 ⇐⇒ s = 2/9Hence x0= (4/9, 4/9, 4/9).Problem 1 Let Π be the plane in R3passingthrough the points (2, 0, 0), (1, 1, 0), and (−3, 0, 2).Let ℓ be the line in R3passing through the point(1, 1, 1) in the direction (2, 2, 2).(v) Find the angle between the line ℓ and the plane Π.Let φ denote the angle between vectors u = (2, 2, 2)and p = (2, 2, 5). Our angle is ψ = |π/2 − φ|.cos φ =u · p|u||p|=18√12√33=3√11ψ =π2− arccos3√11= arcsin3√11Problem 1 Let Π be the plane in R3passingthrough the points (2, 0, 0), (1, 1, 0), and (−3, 0, 2).Let ℓ be the line in R3passing through the point(1, 1, 1) in the direction (2, 2, 2).(vi) Find the distance from the origin to the plane Π.The equation of the plane Π is 2 x + 2y + 5z = 4.Hence the distance from a point (x0, y0, z0) to Πequals|2x0+ 2y0+ 5z0− 4|√22+ 22+ 52=|2x0+ 2y0+ 5z0− 4|√33.The distance from the origin to the plane is equal to4/√33.Bonus Problem 5. Find the volume of thetetrahedron with vertices a = (1, 0, 0),b = (0, 1, 0), c = (0, 0, 1), and d = (2, 3, 5).Vectors x = b − a = (−1, 1, 0), y = c − a = (−1, 0, 1), andz = d − a = (1, 3, 5) are represented by adjacent edges of thetetrahedron.It follows that the volume of the tetrahedron is16x · (y × z).x · (y × z) =−1 1 0−1 0 11 3 5= (−1)0 13 5− 1−1 11 5= 9.Thus the volume of the tetrahedron is16x · (y × z)=16· 9 = 1.5.xyzParallelepiped is a prism.(Volume) = (area of the base) × (height)Area of the base = |y × z|Volume = |x · (y × z)|xyzTetrahedron is a pyramid.(Volume) =13(area of the base) × (height)Area of the base =12|y × z|=⇒ Volume =16|x · (y × z)|Problem 2. Let f (x) = a cos 2x + b cos x + c.Find a, b,