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(3.2) Use the elementary row operations to determine for which value (s) of

`k`

will 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

`C`

with

`C=2A`

. (4.2) Use the data above to find the coordinates of the point

`D`

such that the quadrilateral

`ABCD`

is 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

`U`

passing 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)`

.