Vectors 1 Tutorial Preview
2
www.whitegroupmaths.com Vectors 1 1. Given that a+b is parallel to c and that |a|=3, |b|=2, |c|=1 and that the angle between a and b is ) 4 1 ( cos 1 − , show that either a+b=4c or a+b=-4c. In each case, find the angle between a and c. 2. Given three points A(4,-5,4) , B(4,-3,2) and C (1,-6,2), find the projection vector of BA on BC. Hence find the position vector of (i) the foot of the perpendicular from A to the line BC. (ii) the image of A when it is reflected in the line BC. 3. Given that , b q b p • = • , c q c p • = • and mc lb a q p 2 2 2 + + = + where l and m are constants, show that if b and c are perpendicular, then p c c c c a p c b b b b a p b a q − • • − • + • • − • + = ) ( 2 ) ( 2 2 4. By considering AC AB AD × • , show that the points A(4,5,1), B(-4,4,4), C(0,-1,-1) and D(3,9,4) are coplanar. 5. There are four points A(1,-1,2), B(2,1,0), C(3,-2,3) and D(3,-2,1). (i) Obtain a vector perpendicular to the plane ABC. (ii) Find the length of the projection of DA on the vector found in part (i). (ii) If E is the foot of the perpendicular from D to the plane ABC, find DE and OE. 6. Given three points A(0,1,2) , B(3,2,1) ,C(1,-1,0), find a unit vector perpendicular to the plan ABC. Find also the area of the triangle ABC. 7. A and B are two points given by OA= λ i+5j+5k and OB =5i+ β j, where λ and β are real numbers. C is a point given by (5,5,5). Show that if there exists a plane containing O, A, B, C, then either λ =5 or β =0. Given that β =0, if there exists a plane containing O, A B and C, find the vector perpendicular to this plane. Given that λ =3 and β =3, find the perpendicular distance from C to the plane OAB.
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Transcript of Vectors 1 Tutorial Preview
www.whitegroupmaths.com Vectors 1
1. Given that a+b is parallel to c and that |a|=3, |b|=2, |c|=1 and that the angle between
a and b is )4
1(cos 1− , show that either a+b=4c or a+b=-4c. In each case, find the
angle between a and c.
2. Given three points A(4,-5,4) , B(4,-3,2) and C (1,-6,2), find the projection vector of
BA on BC. Hence find the position vector of
(i) the foot of the perpendicular from A to the line BC.
(ii) the image of A when it is reflected in the line BC.
3. Given that ,bqbp •=• ,cqcp •=• and mclbaqp 222 ++=+ where l and m
are constants, show that if b and c are perpendicular, then
pccc
capcb
bb
bapbaq −
••−•
+•
•−•+= )(2)(22
4. By considering ACABAD ו , show that the points A(4,5,1), B(-4,4,4), C(0,-1,-1)
and D(3,9,4) are coplanar.
5. There are four points A(1,-1,2), B(2,1,0), C(3,-2,3) and D(3,-2,1).
(i) Obtain a vector perpendicular to the plane ABC.
(ii) Find the length of the projection of DA on the vector found in part (i).
(ii) If E is the foot of the perpendicular from D to the plane ABC, find DE and OE.
6. Given three points A(0,1,2) , B(3,2,1) ,C(1,-1,0), find a unit vector perpendicular to
the plan ABC. Find also the area of the triangle ABC.
7. A and B are two points given by OA=λ i+5j+5k and OB =5i+ β j, whereλ and β
are real numbers. C is a point given by (5,5,5). Show that if there exists a plane
containing O, A, B, C, then either λ =5 or β =0.
Given that β =0, if there exists a plane containing O, A B and C, find the vector
perpendicular to this plane.
Given that λ =3 and β =3, find the perpendicular distance from C to the plane
OAB.
www.whitegroupmaths.com
8. Find the foot of the perpendicular from the point Q(5,2,4) to the straight line l
whose vector equation is r=4i+2j+k+α (-2i-j+k).
Hence find the perpendicular distance from the point Q to the line l .
9. Find the position vector of the image of the point (-5,2,-1) under a reflection in the
line joining the points (3,1,3) and (-1,-1,-3).
10. The lines l and m whose equations are r=
−
+
2
5
5
1
2 k
s and r=
−+
1
1
2
6
3
7
t
respectively intersect at the point Q.
(i) Find the value of k and the position vector of Q.
(ii) Find the equation of the image of l when reflected in m.
