Exercise 7.2
Solution :
1. For the arbitrary points A, B, C, D and E, and a single vector which is equivalent to
A: DC +CB
= DB
B: CE +DC
= DE
2. shows a cube. Let p = AB, q = AD and r = AE . Express the vectors representing BD, AC and AG in terms of p, q and r.
Consider the triangle ABD shown in Figure. We note that BD represents the third side of the triangle formed when AD are placed head to tail. Using the triangle law
we find :
AB + BD = AD
=> BD = AD - AB
= q - p
Consider the triangle ADC shown in Figure. We note that AD represents the third side of the triangle formed q and when DC are placed head to tail. Using the triangle law
we find :
AC = AD + DC
= q + p
Consider the triangle BAD , EAD , EAB shown in Figure. Using the triangle law
we find :
triangle BAD : BA + AD = BD
triangle EAD : EA + AD = ED
triangle BAE : BA + AE = BE
So AG = AB + AD + AE
= p + q + r
3.Show that NM = 1/2AB.
So that NM = 1/2AB
4.Consider a rectangle with vertices at E, F, G and H.Suppose EF = p and FG = q. Express each of the vectors EH, GH, FH and GE in terms of p and q.
EH = FG = q , GH = -EF = -q
FH = FG + GH
FH = FG + (-EF)
FH = q-p
GE = GH + HE
= -EF - FG
GE = -q - p
6.A force of 15 N acts at an angle of 65◦ to the x axis.Resolve this force into two forces, one directed alongthe x axis and the other directed along they axis
CARTESIAN COMPONENTS
Exercise 7.3
Solution :
1.P and Q lie in the x--y plane. Find PQ, where P is the point with coordinates (5, 1) and Q is the point withcoordinates (−1, 4). Find |PQ|.
2.A and B lie in the x--y plane. If A is the point (3, 4)and B is the point (1,−5) write down the vectors OA , OB and AB. Find a unit vector in the direction of AB.
3.If a = 4i−j+3k and b = −2i+2j−k, nd unit vectors in the directions of a, b and b − a.
4.If a = 5i−2j, b = 3i−7j and c = −3i+4j, express, in terms of i and j, a+b , a+c , c−b , 3c−4b
a+b = 5i - 2j - 3i + 4j = 8i + 9j
a+c = 5i - 2j -3i +4j = 2i +2j
c-b = -3i + 4j -3i +7j = -6i + 11j
3c-4b = -9i + 12j - 12i + 28j = -21i + 40j
5.Write down a unit vector which is parallel to the line y = 7x−3.
6.Find PQ where P is the point in three-dimensional space with coordinates (4, 1, 3) and Q is the point with coordinates (1, 2, 4). Find the distance between P and Q. Further, nd the position vector of the point dividing PQ in the ratio 1:3.
PQ = (3-2 , 2-1 , 1-2) = (1 , 1 , -1)
so that PQ defferance R not have vector to determined.
8.Calculate the length of each of the links and the position vector of the tip of the robot.
9.Show that the vectors a = i + j and b = −i + j are linearly independent
Exercise 7.5
Solution
1.If a = 3i−7j and b = 2i+4j nd a ·b,b·a,a ·a and b·b.
2.If a = 4i+2j−k,b = 3i−3j+3k and c = 2i−j−k, find
3.Evaluate (−13i − 5j) · (−3i + 4j).
(−13i − 5j)·(−3i + 4j) = (-13 x (-3)) + (-5 x 4) = 19
4.Find the angle between the vectors p = 7i + 3j + 2k and q = 2i−j+k.
5. Find the angle between the vectors 7i + j and 4j − k.
6.Find the angle between the vectors 4i−2j and 3i−3j.
7.If a = 7i+8j and b = 5i find a·Ë†b.
We have b = 5i => b = -5i
a.b = (7)(-5) + (8)(0) = -35
9.Given that p = 2q simplify p · q, (p + 5q) · q and (q−p) ·p.
10. Find the modulus of a = i − j − k, the modulus of b = 2i + j + 2k and the scalar product
a · b. Deduce the angle between a and b.
11.If a = 2i+2j−k and b = 3i−6j+2k, find |a|, |b|, a · b and the angle between a and b.
14.If a = 3i−2j, b = 7i+5j and c = 9i−j, show that a· (b−c) = (a·b)− (a· c).
15.Find the work done by the force F = 3i − j + k in moving an object through a displacement
s = 3i+5j.
16.A force of magnitude 14 N acts in the direction i + j + k upon an object. It causes the object to move from point A(2, 1, 0) to point B(3, 3, 3). Find the work done by the force.
THE SCALAR PRODUCT
Exercise 7.6
Exercise 7.6
Solution
1.Evaluate
2.If a = i−2j+3k and b = 2i−j−k, nd
a) axb
b) bxa
3.If a = i−2j and b = 5i+5k fi nd a×b.
4.If a = i+j−k,b = i−j and c = 2i+k find
a) (axb)xc
b) ax(bxc)
5.If p = 6i+7j−2k and q = 3i−j+4k find |p|, |q| and |p × q|. Deduce the sine of the angle between p
and q.
6.For arbitrary vectors p and q simplify
a) (p+q)×p = (pxp+qxp)=qxp
b) (p+q)×(p−q) = (pxp-pxq+pxq-qxq)=2qxp
7.If c = i+j and d = 2i+k, find a unit vector perpendicular to both c and d. Further, find the sine of the angle between c and d.
and q.
6.For arbitrary vectors p and q simplify
a) (p+q)×p = (pxp+qxp)=qxp
b) (p+q)×(p−q) = (pxp-pxq+pxq-qxq)=2qxp
7.If c = i+j and d = 2i+k, find a unit vector perpendicular to both c and d. Further, find the sine of the angle between c and d.
8.A, B, C are points with coordinates (1, 2, 3), (3, 2, 1) and (−1, 1, 0), respectively. Find a unit vector perpendicular to the plane containing A, B and C.
10.If a = 7i−j+k,b = 3i−j−2k and c = 9i + j − 3k, show that a×(b+c) = (a×b)+(a×c)
11.a) The area, A, of a parallelogram with base b and perpendicular height h is given by A = bh.Show that if the two non-parallel sides of the parallelogram are represented by the vectors a
and b, then the area is also given by A = |a×b|.
b) Find the area of the parallelogram with sides represented by 2i + 3j + k and 3i + j − k.
13.Suppose a force F acts through the point P with position vector r. The moment about the origin, M, of the force is a measure of the turning effect of the force and is given by M = r × F. A force of 4 N acts in the direction i + j + k, and through the point with coordinates (7, 1, 3). Find the moment of the force about the origin.
THE SCALAR PRODUCT
Solution
1. find the norm of a, the norm of b and a · b. Further, find the norm of a − b.
 |
Review Exercise7
Solution
1. Find a·b and a×b when
a) a = 7i−j+k,b = 3i+2j+5k
a) a = 7i−j+k,b = 3i+2j+5k
4. If a = 6i−j+2k and b = 3i−j+3k, nd |a|, |b|, |a × b|. Deduce the sine of the angle between
a and b.
a and b.
5. If a = 7i+9j−3k and b = 2i−4j, nd ˆa, ˆb, d a×b.
6. By combining the scalar and vector products other types of products can be de ned. The triple scalar product for three vectors is de ned as (a × b) · c which is a scalar. If a = 3i−j+2k,b = 2i−2j−k,
c = 3i+j, nd a×b and (a×b) · c. Show that (a×b) · c = a · (b×c).
c = 3i+j, nd a×b and (a×b) · c. Show that (a×b) · c = a · (b×c).
7. The triple vector product is de ned by (a × b) × c. Find the triple vector product of the vectors given in Question 6. Also nd a · c, b · c and verify that (a · c)b− (b· c)a = (a×b)×c Further,
find a × (b × c) and con rm that a×(b×c) 6= (a×b)×c.
11. In a triangle ABC, denote AB by c, AC by b and CB by a. Use the scalar product to prove the cosine rule: a2 = b2 +c2 −2bc cosA.
13. Find the area of the parallelogram with sides represented by 3i + 5j − k and i + 3j − k.
14. Find the angle between the vectors 7i + 2j and i − 3j.
15. Find a unit vector in the direction of the line joining the points (2, 3, 4) and (7, 17, 1).
16. Show that the vectors i − j and −3i − 3j are perpendicular.
17. Find the norm of each of the vectors
18. a) Use the scalar product to nd the value of the scalar μ so that i + j + μk is perpendicular to the vector i + j + k.
19. The points A, B and C have coordinates (2,−1,−2), (4,−1,−3) and (1, 3,−1).
a) Write down the vectors AB and AC.
a) Write down the vectors AB and AC.
b) Using the vector product nd a unit vector which is perpendicular to the plane containing A, B and C.
c) If D is the point with coordinates (3, 0, 1), use the scalar product to nd the perpendicular distance from D to the plane ABC.
20. The condition for vectors a, b and c to be coplanar(i.e. they lie in the same plane) is a · (b × c) = 0.
a) Show that the vectors a = 4i + 5j + 6k,b = 6i−3j−3k and c = −i+2j+2k are not coplanar.
a) Show that the vectors a = 4i + 5j + 6k,b = 6i−3j−3k and c = −i+2j+2k are not coplanar.
b) Given d = −i + 2j + k, nd the value of so that a, b and d are coplanar.
21. Points A and B have position vectors a and b respectively. Show that the position vector of an arbitrary point on the line AB is given by r = a + (b − a) for some scalar . This is the vector equation of the line.
Comments
Post a Comment