Download Math 1300 Geometry review and extension exercises

Math 1300
Geometry review and extension exercises
This is an extra review sheet. These exercises test your basic skill with vectors, linear systems and geometry
and in a few places to use common sense with vectors. Some hints may be found at the end.
1. Determine whether the system has a unique solution, no solution, or infinitely many solutions by sketching
a graph for each equation (do not solve algebraically).
(a)
2x + y=5
x − y=1
(b)
4x − 2y=−1
2x − y= 2
(c)
3x + 2y= 6
−6x − 4y=−12
(d)
2x + y=10
x − y= 1
x + 3y= 9
2. The 2 × 3 system of equations
a1 x + b1 y + c1 z
= d1
a2 x + b2 y + c2 z
= d2
can be represented by two planes. How are these planes related when:
(a) The system has no solution?
(b) The system has infinitely many solutions, involving one parameter?
(c) The system has infinitely many solutions, involving two parameters?
Explain, in terms of geometry, why this system cannot have a unique solution.
3. Find a nonzero vector perpendicular to the vectors (1, 2, 3), (−2, 1, 0) in two different ways:
(a) Obtain two equations by assuming v = (a, b, c) is perpendicular to both vectors, and solve for a, b and
c.
(b) Compute the cross product of the two vectors.
4. Give lines in both vector and scalar parametric forms (and point-normal when working in R2 ), and planes
in both point-normal and general form:
(a) The line through the origin and (3, −5)
(b) The line through (−1, 2) and (2, −3)
(c) The line perpendicular to (2, 3) + t(1, −1) and passing through (0, −1)
(d) The line parallel to (1, 2, 3) + t(3, −1, −2) and passing through (0, 0, 1)
(e) The plane through the origin, (2, −2, 2) and (−1, 2, −3)
(f) The plane through (1, 0, 0), (−2, 3, 4) and (0, 1, 0)
(g) The plane perpendicular to the line (1, 3, 5) + t(0, 1, 2) and passing through (1, 1, 1)
(h) The line perpendicular to the plane (0, 1, 0) + s(1, 0, 1) + t(1, 2, 3) and passing through (1, 5, −5)
5. For each of the lines asked for in question 4, give an equation or system of equations in x, y or in x, y, z
whose graph is the figure named.
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6. Express the line obtained as the intersection of planes x + 2y − 5z = 10 and 2x + y − 3z = 20, in parametric
form.
7. What is the angle, in degrees (round to two decimal places), between the two planes in the last question?
8. Express the line x − y = 3, y − z = 2 in parametric form.
9. What is the angle, in radians, between the vectors i) (1, 2) and (3, 4)? ii) (1, 2, 3) and (4, 5, 6)? Round
your answers to two decimal places.
10. How close does each of the planes in the above problem come to the origin? At what point?
11. What it the distance between the following lines (in 4-space): u = (1, 2, 3, 4) + t(1, 1, −2, 1) and u =
(−1, 0, 1, 2) + s(1, −2, 0, 1)?
12. Given points A(1, 2, 3), B(2, 2, 0), C(1, 1, 3) and D(0, −1, 2), find:
(a) The area of the parallelogram with vertices at 0, A, B, and 0 being adjacent to A and B.
(b) The area of the triangle ABC.
(c) The volume of the parallelopiped, four of whose vertices are A, B, C, D, with D adjacent to each of
the other three.
13. What is the fourth vertex of a parallelogram in R4 where three of the vertices are (1, 1, −1, 2), (1, 3, 1, 4)
and (2, 5, 1, 0)?
HINTS
4b (−1, 2) + t(3, 5); [4c] Lines are perpendicular when their direction vectors are orthogonal [4d] Lines are
parallel when their direction vectors are scalar multiples of each other [4f] find a normal vector; [4g] and
[4h] How must the direction vector for the line be related to the normal to the plane?
5 Eliminate the parameter. Observe that, in 2D, a line is the graph of one equation and in 3D it is the graph
of a system of two equations.
6 Solve as you would do in chapter 1.
7 How is the angle between planes related to the angle between their normal vectors? Remember something
we know about dot product.
9 dot product. You will need to use an inverse cosine function on a calculator to get decimal approximations
for the angles.
10 Think “normals”.
11 Find a vector v perpendicular to both lines. Project a vector with IP on one line and TP on the other onto
v. Draw a diagram to guide you.
12 Draw rough diagram. Make sure you have vectors representing edges emanating from a common vertex.
Use our formulas for volume and area.
13 Draw a diagram (Yes, it’s 4 dimensional but you need only visualize everything in 2 dimensions. Why?)
Represent sides of the parallelogram by vectors.
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