Linear Dependence and Coplanarity
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MCV4U d1+ B – Linear Dependence and Coplanarity Assignment
Answer all questions with full solutions. Make sure your work is legible, even after you have
scanned it, and submit it as a single file.
1. Write an example of each of the following (assuming it is in 3-dimensional space).
a. A point lying on the x-axis.
b. A point lying on the yz plane.
c. A point lying on both the xy and xz planes.
d. A point lying on all three planes.
e. A point lying on none of the three planes, but equidistant from the xz and yz planes.
2. Triangle ABC has vertices A (-1, 1, 3), B (-1, 3, 5) and C (-3, 3, 3). What kind of triangle is ABC
Justify your answer.
3. The points (1, -2, 4), (3, 5, 7) and (4, 6, 8) are three of four vertices of parallelogram ABCD.
Explain why there are three possibilities for the location of the fourth vertex, and find the three
4. The points A (3, -1, z) B (1, 2, 6) and C (x, 8, 14) are collinear. Find the values of x and z.
5. Explain the meaning of direction angles and their relation to direction vectors.
a. What are the direction angles of the vector [2, 4, -3]
b. If a point P lies on the x-axis, what are the direction angles of the position vector
c. Prove that
2 2 2 cos cos cos 1
d. A vector has direction angles a = 75° and ß= 55°
i. Find the value of
ii. Find a vector that has those direction angles
e. Explain why it is not possible for two of a vector’s direction angles to be less than 45°
f. What is the value of
2 2 2 sin sin sin
6. Explain the meanings of the terms linearly dependent and coplanar. Make sure you demonstrate
that you understand the difference between the terms, and the situation in which linear
dependency implies coplanarity.
7. Determine if the vectors [1, -3, 4], [4, 2, -2] and [3, -2, 3] are coplanar.
8. Give examples of sets of three vectors that are
c. Not coplanar
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9. Explain how you would prove if four given points are coplanar. Use your method to determine if
A (3, 1, 4), B (7, -2, 9), C (0, 8, 2) and D (8, 2, 12) are coplanar.
10. Determine if the following vectors are coplanar. Assume that
are not coplanar.
1 1 2 w v v 2 7
2 2 3 w v v 2
3 1 3 w v v – 7
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