Additional Vectors Problems

1(i) A plane 1∏ has vector equation •r ( 2i +j +2k ) = 4.

(a) Find the coordinates of the foot of perpendicular from the origin to the plane.

(b) Determine whether the line l with equation =r 2j +k +λ ( i +2j -2k) lies in the plane.

(ii) A second plane 2∏ contains the line l and is perpendicular to 1∏ . Find the equation of 2∏ in

the form .dnr =•

(iii) A third plane 3∏ is perpendicular to both planes 1∏ and 2∏ , and is at a perpendicular distance

of 3 units from the point (0, 2, 1). Find the coordinates of the possible points of intersection of the

three planes.

2. Given that the two planes 1P and 2P have equations

1

2

3

6

=

•r

and 5

3

6

2

=

−

−•r respectively,

(a) Verify that the line of intersection of these two planes has equation

−

−

+

−

=

42

22

3

7

3

1

λr .

(b) Find the equation of the plane through A ( 1, 3, -7) and which is perpendicular to both planes

1P and 2P .

(c) Find the vector equation of the line through A which bisects the acute angle between the two

planes 1P and 2P .

(d) Using the results obtained in (a) and (c), find the equation of the plane which bisects the acute

angle between the two planes 1P and 2P .

3. The points ,A B , ,C which form the base of a tetrahedron, have position vectors ,a b and c

respectively. Vertex D has the position vector .d Find in terms of ,a b and ,c

(i) The area of the triangle ABC

(ii) a vector perpendicular to ABC

(iii) Show that the volume of the tetrahedron ABCD is ( ) ( ).6

1cbaacbacbd ו−×+×+ו

(The volume of a tetrahedron is equals to ×3

1base area × perpendicular height. )

4. The planes 1∏ and 2∏ , which meet in the line ,l have vector equations

=r 2i +4j +6k 1θ+ (2i +3k) + 1φ (-4j +5k),

=r 2i +4j +6k 2θ+ (3j +k) + 2φ (-i +j +2k),

respectively. Find a vector equation of the line l in the form .tbar +=

Find the vector equation of the plane 3∏ which contains the line l and passes through the point

4i +3j +2k, giving it in Cartesian form. Deduce that the system of equations

32456 −=−− zyx

2435 =+− zyx

40529 =+− zyx

possess an infinite number of solutions.

5. The lines 1l and 2l have equations

)21(:1 µ+=rl i µ2+ j )34( µ+− k, ℜ∈µ

)4(:2 λarl += i )46( λ++ j )92( λ++ k, ℜ∈λ

respectively, where a is a constant.

(a) Find, correct to the nearest degree, the acute angle between 1l and the −x axis.

(b) The point A has position vector 2i -2j +b k. Given that the line 2l passes through the point

A , find the values of a and .b

Hence find

(i) the position vector of the point of intersection between lines 1l and 2l ,

(ii) the position vector of the foot of perpendicular from point A to the line 1l ,

(iii) the vector equation of the line ,3l the reflection of 2l about 1l .

6. Relative to an origin ,O points A and B have position vectors 3i +4j +k and –i +2j respectively.

The line l has vector equation =r (6i +a j) t+ (i +3j +ak), where t is a real parameter and a is a

constant. The line m passes through the point A and is parallel to the line .OB

(i) Find the position vector of the point P on m such that OP is perpendicular to m .

(ii) Show that the two lines l and m have no common point.

(iii) If the acute angle between the line l and the −z axis is ,60o find the exact values of the

constant .a

7. By expanding ),()( cbcb −•− simplify ( ) )(||22 cbcbcb −•−−+ .

Taking bAC = and ,cAB = deduce the cosine formula for triangle .ABC