the coordinates of C (3) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (4)

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Annice Poole
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1 . The line l has equation, = λ r where λ is a scalar parameter. The line l 2 has equation, = µ r where μ is a scalar parameter. Given that l and l 2 meet at the point C, find the coordinates of C The point A is the point on l where λ = 0 and the point B is the point on l 2 where μ =. 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. (5) (Total 2 marks) 2. The line l has vector equation r = λ and the line l 2 has vector equation r = µ where λ and μ are parameters. The lines l and l 2 intersect at the point A and the acute angle between l and l 2 is θ.

2 Write down the coordinates of A. () Find the value of cos θ. The point X lies on l where λ = 4. Find the coordinates of X. () (d) Find the vector AX (e) Hence, or otherwise, show that AX = The point y lies on l 2. Given that the vector YX is perpendicular to l, (f) find the length of AY, giving your answer to 3 significant figures. (Total 2 marks) 3. Relative to a fixed origin O, the point A has position vector (8i + 3j 2k), the point B has position vector (0i + 4j 4k), and the point C has position vector (9i + 9j + 6k). The line l passes through the points A and B. Find a vector equation for the line l. Find CB.

3 Find the size of the acute angle between the line segment CB and the line l, giving your answer in degrees to decimal place. (d) Find the shortest distance from the point C to the line l. The point X lies on l. Given that the vector CX is perpendicular to l, (e) find the area of the triangle CXB, giving your answer to 3 significant figures. (Total 4 marks) 4. With respect to a fixed origin O the lines l and l 2 are given by the equations l : 2 r = 2 + λ l 2 : r = q + µ 2 p 2 where λ and μ are parameters and p and q are constants. Given that l and l 2 are perpendicular, show that q = 3. Given further that l and l 2 intersect, find the value of p, the coordinates of the point of intersection.

4 9 The point A lies on l and has position vector 3. The point C lies on l 2. 3 Given that a circle, with centre C, cuts the line l at the points A and B, (d) find the position vector of B. (Total 3 marks) 5. With respect to a fixed origin O, the lines l and l 2 are given by the equations l : r = ( 9i + 0k) + λ(2i + j k) l 2 : r = (3i + j + 7k) + μ(3i j + 5k) where λ and μ are scalar parameters. Show that l and l 2 meet and find the position vector of their point of intersection. Show that l and l 2 are perpendicular to each other. The point A has position vector 5i + 7j + 3k. Show that A lies on l. () The point B is the image of A after reflection in the line l 2. (d) Find the position vector of B. (Total 2 marks)

5 6. The points A and B have position vectors 2i + 6j k and 3i + 4j + k respectively. The line l passes through the points A and B. Find the vector AB Find a vector equation for the line l. A second line l 2 passes through the origin and is parallel to the vector i + k. The line l meets the line l 2 at the point C. Find the acute angle between l and l 2. (d) Find the position vector of the point C. (Total marks) 7. The line l has equation r = 0 + λ. 0 The line l 2 has equation 2 r = 3 + µ. 6 Show that l and l 2 do not meet.

6 The point A is on l where λ =, and the point B is on l 2 where µ = 2. Find the cosine of the acute angle between AB and l. (Total 0 marks) 8. The point A has position vector a = 2i +2j + k and the point B has position vector b = i + j 4k, relative to an origin O. Find the position vector of the point C, with position vector c, given by c = a + b. () Show that OACB is a rectangle, and find its exact area. The diagonals of the rectangle, AB and OC, meet at the point D. Write down the position vector of the point D. () (d) Find the size of the angle ADC. (Total 4 marks) 9. The point A, with coordinates (0, a, b) lies on the line l, which has equation r = 6i + 9j k + λ(i + 4j 2k). Find the values of a and b. The point P lies on l and is such that OP is perpendicular to l, where O is the origin. Find the position vector of point P.

7 Given that B has coordinates (5, 5, ), show that the points A, P and B are collinear and find the ratio AP : PB. (Total 3 marks) 0. The points A and B have position vectors i j + pk and 7i + qj + 6k respectively, where p and q are constants. The line l, passing through the points A and B, has equation r = 9i + 7j + 7k + 7k + λ(2i + 2j + k), where λ is a parameter. Find the value of p and the value of q. Find a unit vector in the direction of AB. A second line l 2 has vector equation r = 3i + 2j + 3k + µ(2i + j + 2k), where µ is a parameter. Find the cosine of the acute angle between l and l2. (d) Find the coordinates of the point where the two lines meet. (5) (Total 4 marks)

8 . The line l has vector equation r = 8i + 2j + 4k + λ(i + j k), where λ is a parameter. The point A has coordinates (4, 8, a), where a is a constant. The point B has coordinates (b, 3, 3), where b is a constant. Points A and B lie on the line l. Find the values of a and b. Given that the point O is the origin, and that the point P lies on l such that OP is perpendicular to l find the coordinates of P. (5) Hence find the distance OP, giving your answer as a simplified surd. (Total 0 marks) 2. The points A and B have position vectors 5j + k and ci + dj + 2k respectively, where c and d are constants. The line l, through the points A and B, has vector equation r = 5j + k + λ(2i + j + 5k), where λ is a parameter. Find the value of c and the value of d. The point P lies on the line l, and OP is perpendicular to l, where O is the origin. Find the position vector of P. Find the area of triangle OAB, giving your answer to 3 significant figures. (Total 3 marks)

9 3. The line l has vector equation r = λ 4 and the line l 2 has vector equation r = µ, 0 where λ and µ are parameters. The lines l and l 2 intersect at the point B and the acute angle between l and l 2 is θ. Find the coordinates of B. Find the value of cos θ, giving your answer as a simplified fraction. The point A, which lies on l, has position vector a = 3i + j + 2k. The point C, which lies on l 2, has position vector c = 5i j 2k. The point D is such that ABCD is a parallelogram. Show that AB = BC. (d) Find the position vector of the point D. (Total 3 marks) 4. Relative to a fixed origin O, the point A has position vector 5j + 5k and the point B has position vector 3i + 2j k. Find a vector equation of the line L which passes through A and B.

10 The point C lies on the line L and OC is perpendicular to L. Find the position vector of C. (5) The points O, B and A, together with the point D, lie at the vertices of parallelogram OBAD. Find, the position vector of D. (d) Find the area of the parallelogram OBAD. (Total 3 marks) 5. Relative to a fixed origin O, the vector equations of the two lines l and l 2 are l : r = 9i + 2j + 4k + t( 8i 3j + 5k), and l 2 : r = 6i + αj + 0k + s(i 4j + 9k), where α is a constant. The two lines intersect at the point A. Find the value of α. Find the position vector of the point A. () Prove that the acute angle between l and l 2 is 60. (5)

11 Point B lies on l and point C lies on l 2. The triangle ABC is equilateral with sides of length 4 2. (d) Find one of the possible position vectors for the point B and the corresponding position vector for the point C. (Total 6 marks) 6. The equations of the lines l and l 2 are given by l : l 2 : r = i + 3j + 5k + λ(i + 2j k), r = 2i + 3j 4k + µ (2i + j + 4k), where λ and µ are parameters. Show that l and l 2 intersect and find the coordinates of Q, their point of intersection. Show that l is perpendicular to l 2. The point P with x-coordinate 3 lies on the line l and the point R with x-coordinate 4 lies on the line l 2. Find, in its simplest form, the exact area of the triangle PQR. (Total 4 marks) 7. Relative to a fixed origin O, the point A has position vector 3i + 2j k, the point B has position vector 5i + j + k, and the point C has position vector 7i j. Find the cosine of angle ABC. Find the exact value of the area of triangle ABC. The point D has position vector 7i + 3k.

12 (d) Show that AC is perpendicular to CD. Find the ratio AD : DB. (Total marks) 8. Referred to a fixed origin O, the points A and B have position vectors (i + 2j 3k) and (5i 3j) respectively. Find, in vector form, an equation of the line l which passes through A and B. The line l 2 has equation r = (4i 4j + 3k) + µ (i 2j + 2k), where µ is a scalar parameter. Show that A lies on l 2. Find, in degrees, the acute angle between the lines l and l 2. () The point C with position vector (2i k) lies on l 2. (d) Find the shortest distance from C to the line l. 9. The points A, B and C have position vectors 2i + j + k, 5i + 7j + 4k and i j respectively, relative to a fixed origin O. Prove that the points A, B and C lie on a straight line l. The point D has position vector 2i + j 3 k. Find the cosine of the acute angle between l and the line OD. The point E has position vector 3j k. Prove that E lies on l and that OE is perpendicular to OD. (Total marks)

13 20. The line l has vector equation The line l 2 has vector equation 4 r = 5 + λ2, where λ is a parameter r = 4 + µ, where µ is a parameter. 3 5 Show that the lines l and l 2 intersect. Find the coordinates of their point of intersection. Given that θ is the acute angle between l and l 2, Find the value of cos θ. Give your answer in the form k 3, where k is a simplified fraction. (Total 0 marks)

Regent College. Maths Department. Core Mathematics 4. Vectors

Regent College. Maths Department. Core Mathematics 4. Vectors Regent College Maths Department Core Mathematics 4 Vectors Page 1 Vectors By the end of this unit you should be able to find: a unit vector in the direction of a. the distance between two points (x 1,