Vector Geometry

STEP 2006, Math II, #8: Show that the line through the points with position vectors

and has equation

where

is a scalar parameter. [Hint: Try rewriting this as .]

The sides of OA and CB of a trapezium OABC are parallel, and OA > CB. The point E on OA is such that OE : EA = 1 : 2 , and

F is the midpoint of CB. The point D is the intersection of OC produced and AB produced; the point G is the intersection of OB

and EF; and the point H is the intersection of DG produced and OA. Let

and be the position vectors of the points A and C,

respectively, with respect to the origin O.

(i) Show that B has position vector

for some scalar parameter . [Hint: Think too about how big is.]

(ii) Find, in terms of

, and only, the position vectors of D, E, F, G, and H. Determine the ratio OH : HA.

[Hints: find where lines OD and AB intersect. Then keep in mind that if two vectors are not parallel then one cannot be a scalar

multiple of the other. Next figure out the position vector of F. Then find out where lines OB and EF meet. This should be the

position vector of G. Finally, you could parametrize lines OA and DG and thereby find H.]

A complete solution to this problem is posted below.

STEP 2011, Math I, #4: The distinct points P and Q, with coordinates and respectively, lie on the curve

[Hint: Try parametrizing the curve and finding the tangent line using the tangent vector]. The tangents to the curve at

P and Q meet at the point T. Show that T has coordinates

. You may assume that and .

The point F has coordinates and is the angle TFP. Show that

[Hint: Use vectors and ] and deduce that the line FT bisects the angle PFQ.

A complete solution to this problem is posted below.

STEP 2008, Math II, #8: The points A and B have position vectors

and , respectively, relative to the origin O. The points

A, B, and O are not collinear. the point P lies on AB between A and B such that

AP : PB =

Write down the position vector of P in terms of

, , and . [Hint: parametrize the line AB using then check out the ratio

above. Keep in mind that, since this ratio is given,

becomes a constant.]

Given that OP bisects

AOB , determine in terms of and , where and .

[Hint: Use the cosine angle formula to show that

Keep in mind that

and are not parallel.]

The point Q also lies on AB between A and B, and is such that AP = BQ. Prove that

STEP 2006, Math I, #8: Note that the volume of a tetrahedron is equal to times the area of the base times the height.

The points O, A, B, and C have coordinates , , , and , respectively, where , , and are positive.

[Hint: Draw this tetrahedron using the coordinate axes.]

(i) Find, in terms of

, , and , the volume of the tetrahedron OABC.

(ii) Let angle ACB =

. Show that

and find, in terms of

, , and , the area of triangle ABC.

Hence show that

, the perpendicular distance of the origin from triangle ABC, satisfies

[Hint: You already know the volume of the tetrahedron, but you're being asked to find it using a different method and then compare the results.]

STEP 2004, Math II, #6: The vectors

and lie in the plane . Given that and , find, in terms of and , a vector parallel to and a vector

perpendicular to , both lying in the plane , such that

The vector

is not parallel to the plane and is such that and . Given that , find, in terms of , , and , vectors , , and such that

and are parallel to and , respectively, is perpendicular to the plane and

STEP 2012, Math II, #7: Three distinct points, , and with position vectors , and respectively, lie on a circle of radius 1 with its centre at the origin O. The point G has position vector

The line through and G meets the circle again at the point and the points and are defined correspondingly.

Given that , where is a positive scalar show that

and hence that

where

, , and .

Deduce that

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