Vectors in Two Dimensions (Copy)
1.
Write the vector representing the translation that maps A(2,3) to B(7,−1).
2.
Point P(−4,5) is translated by vector (6,−3). Find its image.
3.
Point Q(8,−2) is translated by vector (−5,7). Find the image.
4.
Find the vector from A(3,1) to B(−2,6).
5.
Find the magnitude of the vector (5,12).
6.
Simplify the vector sum (3,−4)+(−7,9).
7.
Simplify the vector difference (8,−3)−(−2,5).
8.
Multiply the vector (4,−6) by scalar 3.
9.
Multiply the vector (−2,7) by scalar −2.
10.
Given a= (2,−1), b=(−3,4), find a+b.
11.
Given a=(5,0), b=(0,−2), find 2a−3b.
12.
Point P(1,2) is mapped to Q(−2,8). Write PQ as a vector.
13.
The position vector of A is (4,−3). The position vector of B is (−1,2). Find vector AB.
14.
The position vector of P is (3,4). Translate by (−5,7). Write new coordinates.
15.
Simplify: (6,−2)+(−4,5)+(3,−7).
16.
If vector a=(7,2), find −a.
17.
Vector b=(−8,6). Find magnitude of b.
18.
Vector u=(4,3). Find ½u.
19.
A translation maps (x,y)→(x+3,y−5). Write its vector.
20.
Point P(0,0) is mapped to Q(−9,12). Find PQ vector and its magnitude.
21.
Simplify 2(3,−4)−(5,−2).
22.
Find the unit vector in the direction of (3,4).
23.
Vector u=(2,5), v=(−1,3). Find u+2v.
24.
If OA=(4,1), OB=(−2,7), find AB.
25.
Simplify (7,−8)−3(2,−5).
26.
If vector PQ=(6,−2), find vector QP.
27.
Find the vector representing the diagonal of a parallelogram with adjacent sides (4,1) and (−2,5).
28.
Show that (2,3) and (−4,−6) are parallel vectors.
29.
Point A(2,−1), B(5,3). Find vector AB and its magnitude.
30.
A triangle has vertices A(0,0), B(3,1), C(1,4). Write vectors AB and AC, then find BC.
1.
Vector = (7−2, −1−3) = (5, −4).
2.
Image = (−4+6, 5−3) = (2, 2).
3.
Image = (8−5, −2+7) = (3, 5).
4.
AB = (−2−3, 6−1) = (−5, 5).
5.
Magnitude = √(5²+12²) = √(25+144) = √169 = 13.
6.
(3, −4)+(−7, 9) = (−4, 5).
7.
(8, −3)−(−2, 5) = (8+2, −3−5) = (10, −8).
8.
3(4, −6) = (12, −18).
9.
−2(−2, 7) = (4, −14).
10.
a+b = (2, −1)+(−3, 4) = (−1, 3).
11.
2a−3b = 2(5,0)−3(0,−2) = (10,0)−(0,−6) = (10,6).
12.
PQ = (−2−1, 8−2) = (−3, 6).
13.
AB = (−1−4, 2−(−3)) = (−5, 5).
14.
Image = (3−5, 4+7) = (−2, 11).
15.
(6, −2)+(−4, 5)+(3, −7) = (5, −4).
16.
−a = (−7, −2).
17.
|b|=√((−8)²+6²)=√(64+36)=√100=10.
18.
½(4,3) = (2,1.5).
19.
Vector = (3, −5).
20.
PQ = (−9,12).
Magnitude = √((−9)²+12²)=√(81+144)=√225=15.
21.
2(3, −4)−(5, −2) = (6, −8)−(5, −2) = (1, −6).
22.
Unit vector = (3/5, 4/5).
23.
u+2v = (2,5)+2(−1,3) = (2−2, 5+6) = (0,11).
24.
AB = OB−OA = (−2,7)−(4,1) = (−6,6).
25.
(7, −8)−3(2, −5) = (7, −8)−(6, −15) = (1, 7).
26.
QP = −PQ = (−6,2).
27.
Diagonal = (4,1)+(−2,5) = (2,6).
28.
(−4,−6) = −2(2,3).
Hence vectors are parallel.
29.
AB = (5−2, 3−(−1)) = (3,4).
Magnitude = √(3²+4²)=√25=5.
30.
AB=(3−0,1−0)=(3,1).
AC=(1−0,4−0)=(1,4).
BC=AC−AB=(1−3,4−1)=(−2,3).