Position vectors

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.

Relative to a fixed origin O

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

• the point B has position vector 2 i + 4 j + a k

where a is a positive integer.

Find the smallest value of a for which

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.

Find the position vector of D.

With respect to a fixed origin O, the line l₁ is given by the equation

where is a scalar parameter. The point P lies on l₁ and has coordinates (2, c, d), where c and d are constants. The point A does not lie on l₁1 DQG KDV FRRUGLQDWHV í

Find the exact value of the distance PA.

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

Write down a vector equation for the line l₂ The acute angle between PA and l₂ is

The point B lies on the line l₂ such that PA = PB.

Find the coordinates of the point B.

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.

With respect to a fixed origin O, the line l₁ is given by the equation

where is a scalar parameter. The point A lies on l₁ where = 1

Find the coordinates of A.

1

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

Write down a vector equation for the line l₂

Find the exact value of the dista AP.

The acute angle between AP and l₂ is .

Find the value of cos

A point E lies on the line l₂ Given that AP = PE,

find the area of triangle APE,

find the coordinates of the two possible positions of E.

The line / has vector equation

where a, b and c are constants and 4 is a parameter. Given that the point A lies on the line /,

find the value of a. (3) Given also that the vector 4B is perpendicular to /

find the distance AB. (2) The image of the point B after reflection in the line / is the point B’.

Find the position vector of the point B’. (2)

With respect to a fixed origin O, the line l has equation

The point A lies on l and has coordinates (3, – 2, 6). The point P has position vector (– p i + 2 p k) relative to O, where p is a constant. Given that vector PA is perpendicular to l,

find the value of p.

Given also that B is a point on l such that BPA = 45°,

find the coordinates of the two possible positions of B.

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

where and are scalar parameters.

Given that l₁ and l₂ meet, find the position vector of their point of intersection.

Find the acute angle between l₁ and l₂, giving your answer in degrees to 1 decimal place.

Given that the point A has position vector 4i + 16j – 3k and that the point P lies on l₁ such that AP is perpendicular to l₁,

find the exact coordinates of P.

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

where and — are scalar parameters.

Show that l₁ and l₂ meet and find the position vector of their point of intersection A.

Find, to the nearest 0.1, the acute angle between l₁ and l₂.

Show that B lies on l₁.

Find the shortest distance from B to the line l₂, giving your answer to 3 significant figures.

position vector 2i 2j k. The points A and B lie on a straight line l.

Find a vector equation of l.

Given that AC is perpendicular to l, find

the value of p,

the distance AC.

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

where λ and μ are parameters and p and q are constants. Given that l₁ and l₂ are perpendicular,

show that q = –3.

Given further that l₁ and l₂ intersect, find

the value of p,

the coordinates of the point of intersection.

Given that a circle, with centre C, cuts the line l₁ at the points A and B,

find the position vector of B.

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

where λ and μ are scalar parameters.

Show that l₁ and l₂ meet and find the position vector of their point of intersection.

Show that l₁ and l₂ are perpendicular to each other.

The point A has position vector 5 i + 7 j + 3 k.

Show that A lies on l₁.

The point B is the image of A after reflection in the line l₂.

Find the position vector of B.