Vector arithmetic

Given that

• PQ → = 2i + 8j – 2k

• QR → = 6i + 6k

find → PR.

Hence show that triangle PQR is both right-angled and isosceles.

Figure 2 shows a sketch of the straight line l.

Line l passes through the points A and B.

Relative to a fixed origin O

• the point A has position vector 2 i – 3 j + 5 k

• the point B has position vector 5 i + 6 j + 8 k

Given that a point P lies on l such that

find the possible position vectors of P.

Figure 1 shows a sketch of triangle ABC. Given that

find AC

show that cosABC =

Figure 1 shows a sketch of triangle ABC.

Given that

find AC

show that cosABC =

Relative to a fixed origin O ● point A has position vector 2i + 5j − 6k ● point B has position vector 3i − 3j − 4k ● point C has position vector 2i − 16j + 4k

Find AB

Show that quadrilateral OABC is a trapezium, giving reasons for your answer.

Figure 7

Figure 7 shows a sketch of triangle OAB. The point C is such that OC = 2OA. The point M is the midpoint of AB. The straight line through C and M cuts OB at the point N. Given OA = a and OB = b

Find CM in terms of a and b

Hence prove that ON: NB = 2: 1

With respect to a fixed origin O, the lines l₁ and l₂ are given by the equations

where λ and are scalar parameters. The lines l₁ and l₂ intersect at the point X.

Find the coordinates of the point X.

Find the size of the acute angle between l₁ and l₂, giving your answer in degrees to 2 decimal places.

2

Find the distance AX, giving your answer as a surd in its simplest form.

The point Y lies on l₂. Given that the vector YA is perpendicular to the line l₁

find the distance YA, giving your answer to one decimal place.

The point B lies on l₁ where |AX | = 2|AB|.

Find the two possible position vectors of B.

Relative to a fixed origin O, the point A has position vector 4

and the point B has position vector 3

The line l₁ passes through the points A and B.

Find the vector AB.

Hence find a vector equation for the line l₁

The point P has position vector 2

Given that angle PBA is ,

The line l₂ passes through the point P and is parallel to the line l₁

Find a vector equation for the line l₂ The points C and D both lie on the line l₂ Given that AB = PC = DP and the x coordinate of C is positive,

find the coordinates of C and the coordinates of D.

find the exact area of the trapezium ABCD, giving your answer as a simplified surd.

Relative to a fixed origin O, the point A has position vector 10 i + 2j + 3k, and the point B has position vector 8 i + 3j + 4k. The line l passes through the points A and B.

Find a vector equation for the line l.

The point C has position vector 3 i + 12j + 3k. The point P lies on l. Given that the vector is perpendicular to l,

find the position vector of the point P.

The points A and B have position vectors 2 i + 6 j – k and 3 i + 4 j + k respectively.

Find the position vector of the point C.