Vector Geometry (Copy)
1.
Write down the position vector of point A(3, −2).
2.
If OA = (4,1) and OB = (−2,5), find AB.
3.
Given OA = (6,2) and OB = (3,5), express vector AB in terms of OA and OB.
4.
If OP = (2,−3), OQ = (5,4), find PQ.
5.
Find the midpoint of AB, where A has position vector (2,4) and B has (6,8).
6.
Find the length of vector AB where A(1,1), B(4,5).
7.
If OA = (3,−1), OB = (9,−3), show that OA and OB are parallel.
8.
Show that A(1,2), B(3,6), and C(5,10) are collinear using vectors.
9.
Point P divides AB in the ratio 2:1, where A(0,0), B(6,9). Find the position vector of P.
10.
If u = (4,3), v = (−2,1), find u+2v.
11.
If u = (5,−2), v = (1,4), find 3u−2v.
12.
Given OA=(2,5), OB=(−4,3), find vector BA.
13.
If A=(2,1), B=(6,5), C=(10,9), show that A, B, C are collinear.
14.
If OA=(7,3), OB=(−2,5), find |AB|.
15.
In triangle OAB, OA=(2,1), OB=(4,5). Find vector AB.
16.
A point divides the line joining A(2,−2) and B(8,4) in the ratio 1:2. Find the position vector of the point.
17.
Find λ if vectors (6,9) and (2λ,3λ) are parallel.
18.
Given OA=(4,7), OB=(10,19). Show that A, B, and O are collinear.
19.
Express OC in terms of OA and AB, where OA=(3,1), AB=(2,5).
20.
If u=(7,2), v=(3,4), write u−v as a column vector.
21.
Find coordinates of the centroid of triangle with vertices A(2,1), B(4,5), C(6,7) using position vectors.
22.
Show that vectors (−8,6) and (4,−3) are parallel or not.
23.
Find the ratio in which P divides AB if OP = (5,5), OA=(2,2), OB=(8,8).
24.
If OA=(1,3), OB=(7,9), show that AB is parallel to OA.
25.
In parallelogram OABC, OA=(3,2), OB=(5,4). Find OC.
26.
In a triangle, OA=(6,4), OB=(2,8). Find vector AB.
27.
If OP=(4,−2), OQ=(−2,1), find vector PQ and its magnitude.
28.
If A=(1,4), B=(3,0), C=(7,−8), show that AB and AC are not parallel.
29.
If OA=(2,−1), OB=(8,2), find the position vector of the point dividing AB in ratio 1:2.
30.
In a triangle, OA=(4,0), OB=(0,6), find vector AB and show the coordinates of midpoint of AB.
1.
Position vector of A(3,−2) is simply (3,−2).
2.
AB = OB − OA = (−2,5) − (4,1) = (−6,4).
3.
AB = OB − OA = (3,5) − (6,2) = (−3,3).
So AB = −OA + OB.
4.
PQ = OQ − OP = (5,4) − (2,−3) = (3,7).
5.
Midpoint = (½(2+6), ½(4+8)) = (4,6).
6.
AB = (4−1, 5−1) = (3,4).
|AB| = √(3²+4²) = 5.
7.
OB = 3(3,−1) = 3OA.
So they are parallel.
8.
AB = (3−1,6−2) = (2,4).
AC = (5−1,10−2) = (4,8).
AC = 2AB → collinear.
9.
P divides AB in ratio 2:1 → OP = (2/3)OB + (1/3)OA.
OA=(0,0), OB=(6,9).
OP = (4,6).
10.
u+2v = (4,3)+2(−2,1) = (4,3)+(−4,2) = (0,5).
11.
3u−2v = 3(5,−2)−2(1,4) = (15,−6)−(2,8) = (13,−14).
12.
BA = OA − OB = (2,5)−(−4,3) = (6,2).
13.
AB = (6−2,5−1) = (4,4).
BC = (10−6,9−5) = (4,4).
Same direction → collinear.
14.
AB = (−2−7,5−3) = (−9,2).
|AB| = √(81+4) = √85 ≈ 9.22.
15.
AB = (4,5)−(2,1) = (2,4).
16.
Position vector = (1/3)(2,−2) + (2/3)(8,4) = (2/3+16/3, −2/3+8/3) = (18/3,6/3) = (6,2).
17.
For (6,9) and (2λ,3λ) to be parallel → 6/2λ = 9/3λ.
=3/λ=3/λ → holds if λ≠0.
So any λ works (non-zero).
18.
OB = (10,19). OA = (4,7).
OB = (2.5)OA → O, A, B are collinear.
19.
OC = OA + AB = (3,1)+(2,5) = (5,6).
20.
u−v = (7,2)−(3,4) = (4,−2).
21.
Centroid = (1/3)((2+4+6),(1+5+7)) = (12/3,13/3) = (4,13/3).
22.
Check ratio: (−8,6) vs (4,−3).
If parallel, must be constant multiple.
(−8)/(4)=−2, 6/(−3)=−2. Same → parallel.
23.
OA=(2,2), OB=(8,8).
AP= (5−2,5−2)=(3,3).
AB=(6,6).
Ratio=AP:PB=1:1 → P is midpoint.
24.
AB = (7−1,9−3) = (6,6).
OA = (1,3).
AB = 2(OA). Parallel.
25.
Parallelogram: OC = OA+OB = (3,2)+(5,4)=(8,6).
26.
AB = (2−6,8−4) = (−4,4).
27.
PQ = (−2−4,1−(−2)) = (−6,3).
|PQ| = √(36+9)=√45=6.71.
28.
AB = (3−1,0−4) = (2,−4).
AC = (7−1,−8−4) = (6,−12).
AC = 3AB. So AB and AC are parallel → correction: they are parallel, not non-parallel.
29.
AB = (8−2,2−(−1))=(6,3).
Point divides 1:2 → OP=OA+(1/3)AB=(2,−1)+(1/3)(6,3)=(2+2,−1+1)=(4,0).
30.
AB = (0−4,6−0) = (−4,6).
Midpoint = ((4+0)/2,(0+6)/2) = (2,3).