Math102-10B Introduction to Algebra: Problem Sheet 2.
Assigned Problems : 1e 1f 2a 3b 3c 4a 5c 9b 10a Hand in Date : Monday 2 August
1. If u= (1,2,−3), v= (2,4,1) and w= (3,0,−1) find a) 2u
b) u+v c) u+v−w
d) 3u−4v+ 2w e* u−2v
f* w−u−2v
2. Using the vectors u,v andwfrom the last question find scalarsa,band cso that a* au+bv+cw= (1,1,1)
b) au+bv+cw= (1,2,3)
c) au+bv+cw= (0,0,1) d) au+bv+cw= (0,0,0) 3. Find the vector parametric equation of the line through each pair of points.
a) (8,−1,2) and (5,−4,8).
b* (1,−2,3) and (−3,2,−1).
c* (2,3,0) and (1,0,1).
d) (1,3,2) and (3,9,6).
4. Find the vector parametric equation of the plane through each set of three points.
a* (1,1,0), (1,0,1) and (0,1,1) b) (1,2,3), (1,1,1) and (2,1,3)
c) (0,1,1), (2,1,1) and (4,1,−3) d) (1,1,1), (2,−4,−2) and (10,2,7).
5. Write cartesian equation(s) for each line a) (x, y, z) = (1,2,1) +λ(2,1,3) b) (x, y, z) = (−1,−3,2) +λ(5,1,2)
c* (x, y, z) = (−1,−1,2) +λ(2,0,1) d) (x, y, z) = (−1,−1,2) +λ(2,0,0) 6. Write each line in vector parametric form
a* x−1 2 = y
3 = 4−z 5 . b) x= 2−y
2 = z 3.
c) x−2
3 =y+ 1 andz = 2 d) x= 2 and z = 2.
7. Find a vector parametric equation for these planes.
a) 2x+y−z = 5 b) x−4y+ 3z = 12
c) x+ 2y = 3 b) z = 5 8. Find Cartesian equations for these planes
a) (x, y, z) = (3,−1,2) +s(1,0,−1) +t(2,1,1).
b) (x, y, z) = (1,−1,1) +λ(1,2,1) +µ(1,0,0).
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9. Find the equation of the line where the planes intersect.
a) x+ 2y+z= 4 and 2x−y+ 3z = 5 b* x−y+z = 1 and 3x−y−z = 2 c) x−y+z = 1 and 2x−2y+ 2z = 5 d) 2x−y+ 2z = 4 and 3x−2y+ 3z = 6 e) 3x−z = 2 and 2x−3y= 1
f ) x+y+ 2z = 2 and 2x+ 2y+ 4z = 4
10. In each case find where the line intersects the plane
a* Line (x, y, z) = (5,1,1) +t(−2,1,7) and planex+ 2y+z = 8.
b) Line (x, y, z) = (1,0,1) +λ(−1,1,3) and plane 3x+−y+z = 3.
c) Line (x, y, z) = (1,1,1) +t(−1,1,2) and planex−y−z = 3.
11. Find the intersection of the three planes
a* x+y+z = 1, x+ 2y+ 3z = 2 and 3x+ 2y+z = 3.
b) x+ 2y+z = 0, x+ 2y+ 3z = 0 and 3x+ 2y+z = 0.
c) 2x+y+z = 3, 3x+ 2y+z = 5 and 4x+ 2y+z = 6.
d) x−2y+z = 1, x+y−2z = 2 and 2x−y−z = 3.
12. Find the equation of the line through (1,1,1) parallel to the line (x, y, z) = (3,0,−1) +λ(2,−1,5).
13. Are the three points (−1,2,0), (3,3,2) and (11,5,6) colinear?
Is there a line that contains them all? If so, find the line.
14. Are the four points (1,3,2), (−2,3,5), (8,3,−5) and (−1,0,1) coplanar?
Is there is plane that contains them all? If so, find the plane.
15. Do the lines (x, y, z) = (1,1,3) +t(2,1,−1) and (x, y, z) = (5,6,4) +s(1,2,1) intersect? If so find their point of intersection.
16. Spherical polar coordinates describe vectors by giving their length r, and two angles; a bearing θ measured in the xy plane anticlockwise from the x-axis;
and an angle of elevation φ measured from the xy plane in the direction of the positive z axis.
a) Draw a diagram illustrating this.
b) Use trigonometry to obtain a formula for the x, y and z coordinates of a vector with length r, bearing θ and angle of elevation φ.
c) Use your formula to find the coordinates of a vector length 12 with bearing 60◦ and angle of elevation 30◦.
d) Write the vector (1,1,1) in spherical polar form.
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