1. Vectors
1.
ABCD is a parallelogram. P, Q are the midpoints of the sides AB and CD
respectively. Show that DP and BQ trisect AC and trisected by AC.
Sol:
Let DP and BQ intersect AC at M and N respectively.
Let a , b , c, d , p, q , m , n be the position vectors of the points A, B, C, D, P, Q, M, N
respectively with respect to some origin O.
□ ABCD is a parallelogram.
AB DC
bacd
...(i)
P is the midpoint of AB
a b
p
2
2p a b
...(ii)
Q is the midpoint of CD
cd
q
2
2q c d
...(iii)
For eliminating b from (i) and (ii), subtract (i) from (ii)
b a 2p
b a c d
_________________
2 a 2p c d
2p d 2a c
2p d 2a c
2 1
2 1
This means that M divides DP internally in the ratio 2 : 1 and M divides CA
internally in the ratio 2 : 1.
For eliminating d from (i) and (iii), add (i) and (iii),
…(iv)
2. c d 2q
c d b a
____________
2c 2q b a
2q b 2c a
2q b 2c a
2 1
2 1
This means that N divides BQ internally in the ratio 2: 1 and N divides AC internally
in the ratio 2 : 1.
From (iv) and (v), if follows that DP and BQ trisect AC and are trisected by AC.
2.
If in a tetrahedron, the two pairs of opposite edges are perpendicular, then show
that the edges in the third pair is also perpendicular.
Sol:
Let O-ABC be a tetrahedron. Then (OA, BC), (OB, CA) and (OC, AB) are the pair of
opposite edges.
Take O as the origin of reference and let a, b and c be the position vectors of the
vertices A, B and C respectively. Then
OA a, OB b, OC c,
AB b a, BC c b and CA a c.
Now, suppose the pair (OA, BC) and (OB, CA) are perpendicular to each other.
Then
3. OA. BC 0, i.e., a. (c b) 0
a. c a. b 0
...(i)
and OB. CA 0, i.e., b. (a c) 0
b. a b. c 0
a. b b. c 0
Adding (i) and (2), we get,
...(ii)
a. c b. c 0
c. b c. a 0
i.e., c. (b a) 0
OC. AB 0
∴ the third pair (OC, AB) is perpendicular.
3.
Show that the line segment joining the midpoints of two sides of a triangle is
parallel to the third side and half the length of that of the third side.
Sol:
Let ABC be a triangle and M and N be the midpoints of the AB and AC.
Then we have to show MN parallel to BC and
1
(MN) . (BC).
2
Let a, b, c, m and n be the position
vectors of A, B, C, M and N respectively. Since M and N are the midpoints of AB and
AC respectively,
4. ab
a c
and n
2
2
MN n m
m
ac ab
2 2
1
(a c a b)
2
1
(c b)
2
1
BC
2
Thus MN is non –zero scalar multiple of BC .
MN is parallel to BC.
seg MN is parallel to seg BC.
1
Also, | MN |
| BC |
2
1
(MN) . (BC).
2
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