Calculus and Vectors- Ex 4. A triangle is defined by three points A(0,1,2), Ex 5. B(1,0,2),...
(Total 16 marks) 9. The vertices of the triangle PQR are defined by the position vectors 3. OQ and OR = OP (a) Find PO, PR (ii) (3) (b) Show that cos RPQ = (7) (c) 0 Find sin RPQ. Hence, find the area of triangle PQR, giving your answer in the form a3 (i) (e)
10. -/3 POINTS LARTRIG10 3.4.044. Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to two decimal places.) (-2, -3), (2, 8), (9,2) • (smallest value) (largest value) -/1 POINTS LARTRIG10 3.4.049. Find u. v, where is the angle between u and v. || || = 90, || || = 250, 0 =
C++ question-classes ? Implement a Triangle class in C++. The triangle is defined by its three side lengths - a, b, and c. The class includes appropriate class constructors and public and private methods that perform the following operations: is_triangle - checks whether the given side lengths form a proper triangle; area - returns the area of the triangle; perimeter - returns the perimeter of the triangle; angle_a, angle_b, angle_c - return the vertex angles (in degrees) opposite to side...
Asvanced Calculus 12. Consider A = R'. Ifu, v E A, the Hamming distance is defined by d(u, v) to be the number of coordinates in which they differ. For example if u = (0,1,2) and v = (0,5,6) then d(u, v) = 2 since the vectors differ in the 2nd and 3rd coordinate, but agree in the 1st. (a) Show that d(u, v) is a metric on A. (b) Let S be the subset of A consisting of the...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Q = (0,6, -4) R= (5,-4, -5) Consider the triangle with vertices: P= (-2,0, -1) (a) Find the vectors PO, PŘ, and QŘ (b) What is the measure of the angle at P (ZQPR)? (c) What is the perimeter of the triangle APQR ? (d) What is the area of the triangle APQR? (e) Find a vector that is perpendicular to the plane containing P, Q, and R Verify that the vector you have found is perpendicular to PO (f)...
Find the three angles of the triangle with the given vertices: P(1,1,1), Q(1,−5,2), and R(−2,2,6) Find a nonzero vector orthogonal to the plane through the points: A=(0,1,−1), B=(0,6,−5), C=(4,−3,−4)
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
vectors (b)) Let a, b and c be three vectors such that a is perpendicular to both b and and Ibl = lel. Showr that the equation of the plane through the three points whose position vectors are g, b and c, is le尸. }blicl+b.ef Hence find the equation of the plane through the points (2,-1,1,).(3,2,-1).(-1,3.2) ㄈ vectors (b)) Let a, b and c be three vectors such that a is perpendicular to both b and and Ibl = lel....
Find, correct to the nearest degree, the three angles of the triangle with the given vertices. A(1, 0, -1), B(4, -4, 0), C(1, 2, 5) CAB = o LABC = o ZBCA = O