University of Waikato Department of Mathematics
MATH102-08B Introduction to Algebra Test 1
If u= (u1, u2, u3) and v= (v1, v2, v3) are two vectors then
u.v=|u||v|cos(θ) =±|uk||v|=u1v1+u2v2+u3v3
|u×v|=|u||v||sin(θ)|=|u⊥||v| ; u×v= (u2v3−u3v2, u3v1−u1v3, u1v2−u2v1) u=u⊥+uk ; uk =
u.v v.v
v Question 1.
(a) The following augmented matrices describe systems of equations with vari- ables x, y, z, and w. In each case interpret the matrix and solve the system.
i.
1 2 0 2 1 0 0 1 3 2 0 0 0 0 0
ii.
1 1 0 0 2 0 0 1 0 3 0 0 0 1 0
iii.
1 1 0 4 0 0 1 1 2 0 0 0 0 0 1
(b) Row reduce the following augmented matrix toreduced echelon form.
0 2 2 1 3 1 1 2 2 3 2 4 6 0 8
(c) Solve the following system of equations.
2y +2z +w = 3
x +y +2z +2w = 3
2x +4y +6z = 8
Question 2.
(a) Find the vector from (3,2,1) to (1,3,5).
(b) What is the distance between the points (3,2,1) and (1,3,5).
(c) Give the equation of the line through (3,2,1) and (1,3,5).
(d) Give the equation of the plane through point (3,2,1) with normal vector (1,3,5).
(e) What is the equation of the line where the planes 3x+ 2y+z = 0 and x+ 3y+ 5z= 0 intersect?
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Question 3. LetP = (3,1,1) , Q= (2,1,2) , R = (1,2,1) (a) Compute P Q~ and P R.~
(b) Compute P Q. ~~ P R and P Q~ ×P R.~ (c) Find the angle RP Q.[
(d) Find the area of the parallelogram spanned by P Q~ and P R~ and hence the area of the triangleP QR.
(e) Find the Cartesian equation of the plane through P, Q and R.
Question 4.
(a) Find the point where the line (x, y, z) = (−1,−2,5)+λ(1,2,−1) intersects the plane 2x−y+ 3z = 9.
(b) Find the angle between the line (x, y, z) = (−1,−2,5) +λ(1,2,−1) and the plane 2x−y+ 3z = 9.
(c) Find the equation of the plane through (2,1,3) which is perpendicular to the line (x, y, z) = (−1,−2,5) +λ(1,2,−1)
(d) Find the distance between the point (2,1,3) and the line (x, y, z) = (−1,−2,5) +λ(1,2,−1).
Question 5. Consider u= (3,2,−2), v= (3,0,−1) and w= (1,−1,0).
(a) Find v⊥ and vk, the components of vperpendicular and parallel to w.
(b) Find the volume of the parallelepiped spanned by u, vand w.
(c) Find the distance between the lines (x, y, z) = (1,0,1) +t(3,2,−2) and (x, y, z) = (0,1,1) +t(3,0,−1).
(d) Show that for any two vectors inR3
(u×v).(u×v) + (u.v)2 = (u.u)(v.v)
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