Review Exercise
1. a. Show that the vectors (2, 3) and (⫺4, 3) may be used as basis vectors
for a plane.
b. Express (3, ⫺1) as a linear combination of (2, 3) and (⫺4, 3).
2. Classify the following sets of vectors as being linearly dependent or linearly
independent. Give reasons for your answers.
a. (3, 5, 6), (6, 10, 12), (⫺3, ⫺5, 6)
b. (5, 1, ⫺1), (6, ⫺5, ⫺2), (3, 8, ⫺2), (⫺40, 39, ⫺29)
c. (7, 8, 9), (0, 0, 0), (3, 8, 6)
d. (7, ⫺8), (14, 19)
e. (0, 1, 0), (0, 0, ⫺7), (7, 0, 0)
b are linearly independent. For what values of t are
3. The vectors ៮៬a and ៮៬
៮៬ ⫺ ៮៬
c៮៬ ⫽ t 2៮៬
a ⫹ ៮៬
b and d៮៬ ⫽ (2t ⫺ 3)(a
b) linearly dependent?
៮៬ ⫺ ៮៬c,
4. If the vectors ៮៬
a, b៮៬, and ៮៬c are linearly independent, show that ៮៬
a ⫺ 2b
៮៬ ⫹ ៮៬
2a
b, and a៮៬ ⫹ ៮៬
b ⫹ ៮៬c are also linearly independent.
5. For each triangle ABC, determine the midpoints of the sides and the
coordinates of the centroid.
a. A(0, 0), B(5, ⫺6), C(2, 0)
b. A(4, 7, 2), B(6, 1, ⫺1), C(0, ⫺1, 4)
៮៬ ⫹ ᎏ2ᎏOP
៮៬ and OM
៮៬ ⫽ ᎏ4ᎏON
៮៬ ⫹ ᎏ1ᎏOQ
៮៬,
6. If ៮៬
OM ⫽ ᎏ35ᎏON
5
5
5
a. in what ratio does P divide NQ?
b. in what ratio does Q divide NM?
7. If M divides AB in the ratio 1:7, show from first principles that
៮៬ ⫽ ᎏ7ᎏOA
៮៬ ⫹ ᎏ1ᎏOB
៮៬.
OM
8
8
8. a. Prove from first principles that the points M, N, and Q are collinear if
៮៬ ⫽ ⫺ᎏ2ᎏ៮៬
ON
OM ⫹ ᎏ191ᎏ៮៬
OQ.
9
ON and ៮៬
OQ.
b. Express ៮៬
OM as a linear combination of ៮៬
230 C H A P T E R 6
9. The point P divides the sides AC of the triangle ABC in the ratio 3:4 and Q
divides AB in the ratio 1:6. Let R be the point of intersection of CQ and BP.
Determine the ratios in which R divides CQ and BP.
10. In the parallelogram ABCD, E divides AB in the ratio 1:4 and F divides BC in
the ratio 3:1. The line segments DE and AF meet at K. In what ratio does K
divide DE, and in what ratio does K divide AF?
11. Prove that the median to the base of an isosceles triangle is perpendicular to
the base.
12. Prove that a line that passes through the centre of a circle and the midpoint
of a chord is perpendicular to the chord.
13. Prove that the medians to the equal sides of an isosceles triangle are equal.
14. Prove that the sum of the squares of the diagonals of a quadrilateral is equal
to twice the sum of the squares of the line segments joining the midpoints
of the opposite sides.
15. In ∆ABC, the points D, E, and F are the midpoints of sides BC, CA, and AB,
respectively. The perpendicular at E to AC meets the perpendicular at F to AB
at the point Q.
៮៬ • (QD
៮៬ ⫺ ᎏ1ᎏ៮៬
a. Prove that AB
AC) ⫽ 0.
2
៮៬ ⫺ ᎏ1ᎏAB
៮៬) ⫽ 0.
b. Prove that ៮៬
AC • (QD
2
៮៬ ⫽ 0.
c. Use parts a and b to prove that ៮៬
CB • QD
d. Explain why these results prove that the perpendicular bisectors of the
sides of a triangle meet at a common point. (Q is called the circumcentre
of the triangle.)
REVIEW EXERCISE
231
Chapter 6 Test
Achievement Category
Questions
Knowledge/Understanding
3, 4
Thinking/Inquiry/Problem Solving
7, 8
Communication
1, 2
Application
5, 6
w are linearly independent. Explain this concept using
1. Three vectors ៮៬
u, v៮៬, and ៮៬
a. an algebraic example
b. a geometric example
2. P divides QR in the ratio 10:⫺3.
៮៬.
a. Express ៮៬
OP as a linear combination of ៮៬
OQ and OR
៮៬.
OP and OQ
b. Express ៮៬
OR as a linear combination of ៮៬
3. a. Copy the three vectors shown in the given diagram onto
graph paper and draw ៮៬c accurately as a linear
combination of ៮៬
a and b៮៬.
៮៬.
b. Determine values of r and s where ៮៬c ⫽ ra៮៬ ⫹ sb
a
b
c
4. The vectors ៮៬
u, v៮៬, and ៮៬
w are coplanar and have magnitudes 5, 12, and 18
៮៬
respectively. u lies between v៮៬ and ៮៬
w, making an angle of 35º with v៮៬ and 20º
៮៬.
with ៮៬
w. Express ៮៬
u as a linear combination of ៮៬v and w
5. F divides AP in the ratio 13:⫺8 and F divides PG in the ratio 4:⫺3.
a. Draw a division-point diagram showing the relative positions of the
four points.
b. In what ratio does P divide AG?
CHAPTER 6 TEST
233
6. a. Form two point-to-point vectors out of the three points A(⫺4, 2, ⫺8),
B(⫺1, ⫺4, ⫺2) and P(1, ⫺8, 2). Demonstrate that the two vectors
are collinear.
b. Express ៮៬
OP as a linear combination of ៮៬
OA and ៮៬
OB.
7. ABCD is a quadrilateral. P, Q, R, and S are the midpoints of its sides.
Prove using vectors that PQRS is a parallelogram.
៮៬ ⫽ kBC
៮៬. Prove that
8. In ∆ABC, D lies on AB and E lies on AC such that DE
៮៬ ⫽ kAB
៮៬ and ៮៬
៮៬, using the fact that ៮៬
AD
AE ⫽ kAC
AB and ៮៬
AC are linearly
independent.
234 C H A P T E R 6