(Solved): (3.2) Use the elementary row operations to determine for which value (s) of k will the matrix below ...
(3.2) Use the elementary row operations to determine for which value (s) of
kwill the matrix below (6) be singular.
A=[[2,2,1,-k],[-1,1,2,1],[1,-2,k,0],[0,1,-1,1]]Question 4: 12 Marks (4.1) Find the area of the triangle with the given vertices
A(1,3),B(-3,5), and
Cwith
C=2A. (4.2) Use the data above to find the coordinates of the point
Dsuch that the quadrilateral
ABCDis a parallelogram. 31 (4.3) Sketch the triangle with vertices
A(1,2),B(3,5), and
C(2,4)and the points
D(0,2),E(0,5)and
F(0,4)on the same
xy-plane. Use the determinant to find the area of the triangle. (4.4) Consider the points
(1,3,-1)and
(2,1,-5). Evaluate the distance between the two points. Question 5: 3 Marks Consider the vectors
vec(u)=(:-2,2,-3:),vec(v)=(:-1,3,-4:),vec(w)=(:2,-6,2:). Evaluate (5.1)
||-(2(vec(u))-3(vec(v)))+(1)/(2)(vec(w))||. (5.2) The unit vector in the direction of
vec(w). Question 6: 2 Marks We assume given a plane
Upassing by the tip of the vectors
vec(u)=<-1,1,2:and
vec(w)=(:1,1,3:). (6.1) Find the dot products
vec(u)*vec(v)and
vec(w)*vec(v)(6.2) Determine whether or not there is a vector
vec(n)that is perpendicular to
U. If yes, then find the vector
vec(n). Otherwise explain why such a vector does not exist? Question 7: 3 Marks Prove that the dot product between two vectors in 3-D is commutative and not associative Question 8: 4 Marks Determine whether
vec(u)and
vec(v)are orthogonal vectors, make an acute or obtuse angle: (8.1)
vec(u)=(:1,3,-2:),vec(v)=(:-5,3,2:). (8.2)
vec(u)=<1,-2,4:. Question 9: 4 Marks Determine proj
vec(a)vec(u)the orthogonal projection of
vec(u)on
vec(a)and deduce
||proj
vec(a)vec(u)||for (9.1)
vec(u)=(:-2,1,-3:),vec(a)=(:-2,1,2:). Also find orthogonal complement and the orthogonal component of
vec(u)along
vec(a).
