Scalar product

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.

With respect to a fixed origin O, the lines l₁ and l₂ are given by the equations 5 0 8 3

where λ and are scalar parameters and p is a constant. The lines l₁ and l₂ intersect at the point A.

Find the coordinates of A.

Find the value of the constant p.

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

The point B lies on l₂ where = 1

Find the shortest distance from the point B to the line l₁, giving your answer to 3 significant figures.

With respect to a fixed origin, the point A with position vector i + 2 j + 3 k lies on the line l₁ with equation 1 0

and the point B with position vector 4i + pj + 3k, where p is a constant, lies on the line l₂ with equation 7 3

Find the value of the constant p.

Show that l₁ and l₂ intersect and find the position vector of their point of intersection, C.

Find the size of the angle ACB, giving your answer in degrees to 3 significant figures.

Find the area of the triangle ABC, giving your answer to 3 significant figures.

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.

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.

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.

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.

Given that l₁ and l₂ meet at the point C, find

the coordinates of C.

The point A is the point on l₁ where λ = 0 and the point B is the point on l₂ where μ = –1.

Find the size of the angle ACB. Give your answer in degrees to 2 decimal places.

Hence, or otherwise, find the area of the triangle ABC.

The line l₁ has vector equation

and the line l₂ has vector equation

where λ and μ are parameters. The lines l₁ and l₂ intersect at the point A and the acute angle between l₁ and l₂ is .

Write down the coordinates of A.

Find the value of cos .

The point X lies on l₁ where λ = 4.

Find the coordinates of X.

Hence, or otherwise, show that = 4√26.4 26.

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

find the length of AY, giving your answer to 3 significant figures.

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.

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.