Al Mahdi Schools
Name: . . . . . . . .
Mathematics
10th-Grade
" Vectors & Vector Coordinates"
W.S-5.
I- Answer with justification by True or False.
1. If AC BD
then, AB CD .
2. If AC BD then, AC BD .
3. If ABCD is a rectangle such that AB 6cm & AD 4cm then, CB CD 2 13cm .
4. If AB k CD where k Z , then :
a. Vectors AB and CD are of the same sense.
b. Vectors AB and CD have same direction.
c.
AB k CD .
5. If A, B & C are any three non-collinear point, so that n 3 AB 2 AC and s 3 AB 2 BC
then the coordinates of n & s in the system A; AB , AC are s 2,1 & n 3,2
6. If ABC is a right isosceles triangle at A , so that AB 6cm , and the point J is defined by
1 1
3
JB AC , then the coordinates of J in the system O; AB , AC are J 3,5
3
2
3
7. In the system (O, i, j ) :
a. The two vectors: u i j , v i j represent basis.
b. If the vectors a 3,2m 1 & s i 2m j are collinear, then m 2
II- Consider in a given plane the two distinct points A & B and a point G defined by the vector
relation: GB 2 AG 0 .
a. Construct the point G .
b. If N is any point of the given plane, then express: 2 NA NB as a function of NG .
c. Determine the locus of the set of points N in the plane such that: 2 NA NB 3 NA
10th-Grade.
Mathematics. W.S-5 Vectors & Vector Coordinates
Page 1 of 3
III- The adjacent figure represents a parallelogram ABCD.
a. Construct the points E & F such that AB 3BE & DF 3 AD.
b. 1) Calculate a & b so that, CF a AB b AD.
2) Calculate m & n so that, EC m AB n AD.
3) Deduce that the points E, C & F are collinear.
c. Consider the reference frame defined by ( A, i, j ), where i AB & j AD.
i. Find coordinates of all points in the given plane.
ii. Deduce using coordinates that the points E, C & F are collinear
1
1
IV- Let ABC be any triangle and x Z so that AE AB x AC and AM x AB AC .
3
3
a. Find EF in terms of BC & x .
b. For what values of x , the vectors EF and BC have the same sense?
c. Calculate the numerical value of x , such that BCFE is a parallelogram.
V- Consider the point G , the centroid of triangle ABC , and the point M to be any point in the
plane of the given triangle.
a. Show that: MA MB MC 3 MG . .
b. Consider the vector: U 3 MA 2 MB 5 MC .
i. Express vector U independent of M .
ii. Find the set of points M for MA MB MC 6 units .
c. Let D be a point defined by AD 2 AB 3 AC .
i. Show that BD k BC , where k is a real number to be determined.
ii. Deduce that the points B, D & C are collinear.
VI- Let ABC be any triangle and D be a point defined by: AD 3 AB 2 AC.
a. Prove that the points B, C & D are collinear, then place D .
b. Take E & F to be any two points defined by: AE AC 2 AB and AF 5 AB 8 AC .
Express the vectors DE then DF in terms of AB & AC.
c. Verify that DF 2 DE 0.
10th-Grade.
Mathematics. W.S-5 Vectors & Vector Coordinates
Page 2 of 3
VII- Consider the plane of reference (O, i, j ) the vectors V 2i 3 j and the points
A(1; 2), B(3; 4), C (1;2) and the point M such that OM xi y j .
1) Calculate the coordinates of vector U 2OB 3OC 3BA .
2) Find a relation between x & y so that the point A, B & M are collinear.
3) Determine the real values of x & y where, AM 2V .
4) Find the coordinates of the points A, B & M in the system (C , i, j ) .
VIII- Consider the plane of reference (o, i, j ) the points A, B, C , D & M such
that: OA i j , OB 4i, OC 4(i j ), OD i 2 j & OM xi y j.
a. Show that ABCD is a parallelogram.
b. Calculate the coordinates of the point E so that, AE 2 AB .
c. Find the coordinates of the point F so that BCFD is a parallelogram.
d. Show that the points are collinear.
e. In the new reference ( A, i j , i 2 j ) the coordinates of the point M becomes M ( X ;Y ) .
1) Find a relation between coordinates of M in the two reference frames.
2) Deduce the coordinates of the points A, B, C & D in the new system.
Chapter
CH-7: Vectors
10th-Grade.
Mastering problems
Exercises
1, 2 & 4
6
9
15 & 16
18 & 19
Mathematics. W.S-5 Vectors & Vector Coordinates
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